diff --git "a/-NAzT4oBgHgl3EQfg_yi/content/tmp_files/load_file.txt" "b/-NAzT4oBgHgl3EQfg_yi/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/-NAzT4oBgHgl3EQfg_yi/content/tmp_files/load_file.txt" @@ -0,0 +1,1843 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf,len=1842 +page_content='Modeling communication asymmetry and algorithmic personalization in online social networks Franco Galantea,∗, Luca Vassioa, Michele Garettob, Emilio Leonardia aPolitecnico di Torino, Corso Duca Degli Abruzzi, 24, 10129 Torino bUniversit`a degli Studi di Torino, Corso Svizzera 185, 10149 Torino Abstract Modeling social interactions and their impact on opinion dynamics has attracted growing interest in recent decades, fuelled by the mounting popularity of online social networks (OSNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' On online social plat- forms, a few individuals, commonly referred to as influencers, produce the majority of content consumed by users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' However, classic opinion models do not capture this communication asymmetry in OSNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We develop an opinion model inspired by observations on leading social media platforms and tailored to the peculiarities of online interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Our work has two main objectives: first, to describe the inherent communication asym- metry in OSNs, where a tiny group of influencers hegemonizes the landscape of social debate, and second, to model the personalization of content by the social media platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We derive a Fokker-Planck equation for the temporal evolution of users’ opinion distribution and analytically characterize the stationary system behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Analytical results, confirmed by Monte Carlo simulations, show how content personalization tends to radicalize user opinion and favor structurally advantaged influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' These emerging behaviors suggest that algorithmic bias, inherently associated with platform filtering, can lead to undesirable outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' As an example application, we apply our model to Facebook during the Italian government crisis in the summer of 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Our work provides a flexible framework to assess the impact of algorithmic filtering on the opinion formation process and a fine-grained tool to study the complex interaction between influencers and social network users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Keywords: opinion dynamics, online social networks, algorithmic personalization, Fokker-Planck equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Introduction In recent years, the way people communicate has changed dramatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' With the advent of the Internet, new communication channels have emerged that allow people to transcend geographical and language barriers thanks to a global communication network and alternatives to text-only interaction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', images and videos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Online social networks (OSNs) are probably the most notable example of such new interaction mechanisms and have greatly influenced our society by fostering discussions and disseminating information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The means that enable online social interactions are profoundly different from traditional interpersonal interactions and other mass media such as newspapers or television.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The amount of content produced on such social media platforms is immense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Therefore, to keep users engaged within the social network, the platform performs filtering to select the posts offered to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This filtering mechanism can reinforce the natural tendency (usually referred to as homophily in the literature) to interact with like-minded people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' And, in turn, can lead to the formation of echo chambers [1], where people who share a similar point of view interact with each other but are isolated from the rest of the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In addition, the reach of certain social media users can be extraordinary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Posts by very influential people can reach a large audience in virtually no time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Another ∗Corresponding author Email addresses: franco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='galante@polito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='it (Franco Galante), luca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='vassio@polito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='it (Luca Vassio), michele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='garetto@unito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='it (Michele Garetto), emilio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='leonardi@polito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='it (Emilio Leonardi) Preprint submitted to Online Social Networks and Media (OSNEM) January 5, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='01478v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='SI] 4 Jan 2023 aspect worth mentioning is the diversity of topics discussed on the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' It ranges from commentaries on the latest sporting events to debates on sensitive issues such as vaccinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Last, communication on OSN takes place asymmetrically, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', few well-known individuals can exert influence on a large audience which, in turn, is composed of far less known people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' These are some crucial aspects that characterize online social networks and distinguish them from ”offline” interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We believe that models seeking to capture the complexity of interactions occurring in online social networks must account for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Trying to understand the mechanisms behind the opinion-forming process is a daunting challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The complexity driving this process is still poorly understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Moreover, the individual reaction to external stimuli is utterly subjective and thus difficult to model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In a continuous framework, opinions can be in- terpreted as a person’s level of agreement with a statement or the interest they show in an issue (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', adoption of technology, politics) and mapped to real-valued intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' It is clear that measuring opinions in such intervals is arbitrary and can only lead to qualitative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Indeed, most of the literature proposes theoretical models without the claim of accurately representing real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Very few works in the literature attempt to validate the emergent behavior of the model with physically observed phenomena (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', [2] [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Some other works take the approach of supporting modeling decisions with real-world observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' [3] did this by interviewing a group of people on specific topics, while Xiong and Liu [4] extracted information from Twitter networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Following the above works, we present a model whose hypotheses are supported by data from social networks and whose outcomes are compared with emerging phenomena on two popular OSNs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', Facebook and Instagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' As the limitations mentioned above also apply to our model, the results discussed here do not aim to be predictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' However, the proposed model provides a tool to study the emerging behavior on online social networks and the impact of algorithmic personalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The main objective of this work is to develop an analytical framework tailored to online interactions, incorporating the following aspects: The asymmetry typically found in OSNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' There exists a relatively small percentage of users of online social networks whose number of followers is orders of magnitude larger than that of other users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' These individuals are commonly referred to in the literature as influencers or opinion leaders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' They are particularly relevant, as their opinions can reach a vast fraction of the social network’s population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The filtering performed on the content by the social media platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Algorithmic personalization appears necessary in the context of OSNs, as the number of daily produced posts has become enormous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The aim is to increase engagement by showing users only the most relevant posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The loop is then closed by taking into account user feedback on the posts received (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', likes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The proposed model: Provides a tool for assessing the impact of different algorithmic personalization policies, focusing on the opinion leaders in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' It can evaluate the extent to which these strategies might hinder diversity of opinion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Exploits the observed characteristics of a large ensemble of Italian influencers from Facebook and Instagram social networks to ground its main hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Allows for comparing its emergent behavior with observations on real online social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Further- more, the explanatory capabilities of the model are used in the study of the opposition of two Italian politicians during a government crisis by identifying a state of public opinion that can lead to the same behaviors observed in the collected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' One peculiar feature of our approach is the concept of reference direction, which is the individual’s main topic of interest and expertise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' To our knowledge, the existing literature has not yet considered the impact of a reference topic for each influencer on the opinion formation process in multi-dimensional spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The influence exerted on non-reference directions depends heavily on platform personalization, which usually depends on how well-known an influencer is in its main field of expertise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' For example, famous public figures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', athletes, models) can express their point of view on potentially sensitive matters and 2 may resonate more than experts due to their popularity in their field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Therefore, since influencers discuss different topics, the reference direction loosely couples seemingly unrelated subjects brought up by the same person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Incorporating a personalization process into the dynamic behavior of the model is another crucial feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The analytical results and the model simulation show that algorithmic personalization favors structurally advantaged individuals, resulting in less diversity of opinion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' It is also interesting to observe that the model undergoes a phase transition in its behavior as a function of the degree of polarisation, at least in the case of two competing influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Below a certain threshold, there are diverse opinions in the population, and above this threshold, one of the influencers tends to polarise users’ attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Section 2 discusses the relevant work in the literature and sets out the rationale for the need for a new opinion model tailored to OSNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The Communication Asymmetry model is presented in Section 3, along with the notation used throughout the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Section 4 presents some observations from real social networks supporting our modeling assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Section 5 is devoted to the mean-field analysis of the model as the number of users grows large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The theoretical results on the steady-state behavior of the model are proved in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Section 6 then investigates the impact of the model parameters on a reference scenario with two influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Analytical findings are validated in section 7 by comparing the results of Monte-Carlo simulations with theoretical predictions specific to our reference scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Section 8 further validates our model with real data collected on Instagram and Facebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' At last, Section 9 concludes the article with a discussion of the implications and limitations of our work, setting the ground for future extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Related work The first steps in the field of opinion dynamics were taken in the late 1950s by a number of social psychologists, among which Solomon Ash [5], John R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' French [6], and Leon Festinger [7] had great resonance in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Ash empirically observed that the individuals he studied engaged in conformist behavior because of the social pressure exerted by the rest of the social group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In short, Ash observed that an individual states a truth about something that is not true (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' “white is black” [3]) when the social group to which the individual belongs asserts it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' French [6] developed a model to capture influence through interpersonal relationships within a group, focusing on leadership and using directed graphs to model interpersonal relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Festinger developed the theory of social comparison, according to which individuals tend to evaluate their position by comparing it with others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Moreover, the tendency to do so decreases the greater becomes the difference in opinion [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Opinion models are divided customarily into two broad classes: those in which opinions are continuous variables and those in which opinions are discrete (often binary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In a recent review [8] examining agent- based opinion models, the authors show that more than 80% of the models considered are continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Much of the seminal work in the field of opinion dynamics is continuous in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' For example, the DeGroot model [9] considers a networked social system in which individuals interact with their neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Individuals average their current opinion with the opinion of their neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The idea behind the model is to describe the process leading to consensus within a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Subsequently, Friedkin and Johnsen [10] extended it by developing a flexible framework from which various opinion models (including French [6] and De Groot [9]) can be derived as particular cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Their model of social influence encompasses both the processes of social conformity and social conflict that lead to behavior that goes beyond simple consensus and represents the persistent disagreement often observed in social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In the early 2000s, Hegselmann and Krause [11] and Deffuant and Weisbuch [12] proposed two similar models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In analogy with the DeGroot model, individuals interact by averaging opinions, but the authors introduced the central idea of bounded confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' According to bounded confidence, individuals interact in a social network with other peers only if their beliefs are not too different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This mechanism implements the concept of homophily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Lorenz, in his review [13], provides the agent-based and density-based formulation of bounded confidence, distinguishing two main models: in the Hegselmann-Krause (HK) model, individuals modify their opinion as a result of interactions with all agents in their neighborhood, whereas, in the Deffuant-Weisbuch model, interactions are pairwise between connected individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 3 In addition to continuous models, discrete models have also appeared in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The first and probably most prominent model of this kind is the voter model, independently introduced by Clifford and Sudbury [14] and Holley and Liggett [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Here, individuals are agents in a network of interactions, holding a binary opinion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' At times dictated by a Poisson clock, an individual adopts the belief of a randomly chosen neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This type of model has attracted a great deal of attention: several extensions have populated the recent literature, for example, taking evolving networks into account [16] [17] or allowing individuals to hold more than one opinion at a time [18], or introducing spontaneous changes of opinion [19] (noisy voter model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' A consistent bulk of research on opinion dynamics comes from the physics literature, among which early contributions are Ben-Naim [20] and Toscani [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The idea underlying these models is that of describing interacting individuals using statistical mechanics by adequately defining the microscopic interactions be- tween the individuals, much like particles in a gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Then, collective statistical phenomena are sought for the overall opinion of the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In the papers mentioned above, Ben-Naim and Toscani consider two mechanisms of opinion formation: compromise, the human tendency to reach a reasonable trade-off on an issue to avoid conflict, and a process of introspection (in other models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', [19], modeled as noise), which the authors believe represents the impact of external sources of information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', media).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' A statistical approach is generally employed to study spin systems, and models such as the Ising model have also been applied to the opinion formation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' An extension of the Ising model is the Sznajd model [22], which implements social validation and for which Slanina and Lavicka [23] derived analytical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In this model, the agreement of individual pairs leads to their neighbors agreeing with them, and a line graph is considered to capture the connection network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' For a comprehensive review of opinion models, we refer to the survey by Castellano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Models tailored to online platforms Most of the seminal literature on opinion dynamics is suited to describe the decision-making process in small groups of individuals, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', a board of directors, or to capture rather regular patterns determined by the daily personal interactions of individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Models such as the voter model have been studied extensively on regular lattices [25] [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The structure of interactions, especially those online, is far from homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' As mentioned earlier, an inherent asymmetry in communication exists in OSNs where a limited number of individuals (influencers) monopolize the discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The voter model has been studied over heterogeneous networks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', [27] [28]) to account for this diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' On such networks, there can exist hubs (strongly connected nodes) playing a role similar to influencers in our framework, although the authors did not explicitly make such a distinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Other works have divided the population into classes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', [29] introduced stubborn agents, and if such individuals have opposing opinions, they hinder the possibility of the population converging to consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Recent work is drawing further attention to online platforms by adapting classical frameworks to the specificities of online interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Valensise et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' [30] have developed an opinion model that embodies al- gorithmic personalization, comparing its behavior to phenomena observed in social networks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', Facebook and Twitter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Our work is different because we consider distinct classes of users, characterizing specifically influencers and closing the interaction loop between users and the platform by a feedback function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Other works that address content filtering bias in social media platforms include [31] [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Peralta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' [31] develop a flexible framework to incorporate algorithmic bias into binary opinion dynamics by having agents interact at a lower rate with individuals who hold an opposing viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Considering both pairwise and group-wise interactions, the authors found that algorithmic bias either leads users to polarize their opinion (in the case of pairwise interactions) or results in the coexistence of beliefs (in the case of group-wise interactions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Validation of opinion formation models In [33], models of opinion dynamics are referred to as idealized because, in most cases, they assume basic underlying principles of interaction and observe emergent social behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' There are two main approaches to validating opinion models in the literature: first, the use of observational data [2][1][30] and second, the use of controlled sociological experiments[34][35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We will focus more on the first portion of the literature, 4 which is more relevant to our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Attempts to validate opinion models are scarce for several reasons: i) the mapping of opinions into values, ii) an adequate definition of links between agents, and iv) the change in opinion after an interaction is hardly measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' A notable exception is election and polling data, which make it possible to attribute a person’s opinion to the political orientation of the chosen candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Fortunato and Castellano [2] have shown that the distribution of vote counts is a universal scaling function and have derived a simple tree-like interaction structure with candidates as roots and an interaction that can turn the individuals reached into “activists” who can spread the idea and convince other individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The results of the model are in good agreement with empirical evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In [36], a noisy voter model could fit data from US elections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Other recent approaches [37] have used shared news on Facebook to assess the extent to which individuals are exposed to opposing views through their (online) friendship relationships, using users’ self-reported ideological affiliations to infer opinion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' They found that individuals have access to cross-cutting content and that the degree of this exposure depends on the composition of one’s friends on social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' A more recent body of literature [1] [30] has directly employed data from online social networks, such as Gab, Facebook, Reddit, and Twitter, to observe the emergence of echo chambers [1] and to validate a model encompassing algorithmic personalization in the process of opinion formation [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The Communication Asymmetry opinion model In this section, we first establish the notation used throughout the paper and then present the Commu- nication Asymmetry (CA) model in its most general formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We conclude the section with a discussion of the strengths and limitations of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Notation In this work, we adopt the following vectorial notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We denote vectors by bold symbols, whereas we denote their components with normal-font symbols whose subscript is the index in the vector, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', a = {ak}k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Lowercase letters denote parameters and dynamical variables associated with an individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In general, index i runs over the set of influencers while index u runs over that of regular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' For those parameters/variables that can be associated with individuals of both classes (either influencers or regular users), the above indices are indicated between superscript parentheses, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', a(i), a(u), to immediately identify the class to which the individual belongs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' If necessary, the dependence of variables on other system parameters is made explicit by specifying the independent variables between parentheses, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', α(·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Italic capital letters denote sets, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', I is the set of all influencers in the population, while |I| is its cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Capital letters represent outcomes of stochastic experiments whose characteristic parameters are low- ercase letters: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', Ω (ω(·, ·)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The operator E[·] represents an expected value, and a bar over a variable, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', ¯a, represents its average value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Whenever we need to express the probability of an event, we use the notation Pr[·].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We employ 1{·} for the indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Lastly, time is denoted by t if it is considered continuous and by n if it is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Description of the model We propose a continuous opinion model with two interacting classes of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Specifically, the population consists of Nu = |U| regular users and Ni = |I| influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This division mimics what happens in real social networks, where a small portion of the population, the influencers, has a much larger number of people following their posts on the online social network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We assume that the generation of new posts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', messages in the OSN, is a Poisson Point Process (PPP) with intensity λ, where each event of the PPP corresponds to the creation of a new post from an influencer i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The corresponding embedded discrete time will be denoted by the integer n ∈ N+, n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', where n is the n-th post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Each post is sent1 to a subset of regular users, identified by the social platform according to an algorithmic personalization described by function ω (to be specified later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Regular users react to these posts through a feedback function θ (specified 5 User Influencer popularity update post generation post provision feedback Platform θ ω Figure 1: The picture depicts the relationship among the three players of the model: regular users, influencers, and the social media platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Moreover, it highlights the role of the feedback function θ provided by the users and the filtering function ω used by the platform to propose the posts to the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Figure 1 highlights this closed loop behavior brokered by the social media platform, positioned between regular users and influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The opinion space is X ⊂ Rd, where each dimension represents an uncorrelated topic on which users have a belief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Therefore, an opinion consists of a d-dimensional vector x(u)(n) ∈ Rd, which evolves as a result of the interaction between a regular user and every influencer on every possible topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This model neglects interactions between regular users2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The prejudice of a user, denoted by z(u), is the other parameter that enters the opinion update rule alongside the user’s current opinion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' It represents the user’s natural inclination toward different topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Unless otherwise specified, we will assume that the user’s initial opinion is set equal to the prejudice: x(u)(0) = z(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We will consider different distributions for the agents’ prejudice over the opinion space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In particular, delta, uniform, and Beta distributions are usually employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Influencers are considered stubborn agents, which means that their opinions do not change over time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', x(i)(n) = x(i)(0) = x(i) = z(i) ∈ Rd, ∀n > 0 and i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' As we will show in Section 4, each influencer has a main topic of interest on which it publishes the majority of its posts and which typically coincides with the topic it is mainly known for on the OSN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' It represents the reference direction r(i) ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='., d − 1} of the influencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Another parameter characterizing influencer i is its consistency c(i)(n), which indicates the probability that such an influencer publishes a post on its reference direction (it might change over time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Note that individuals with high consistency prefer to post in their reference topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Moreover, we denote by f (i) the probability that a post is generated by influencer i at any time instant n, with � i∈I f (i) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' At last, we introduce the popularity vector p(n) := {pi(n)}i∈I, containing the current popularity of all influencers at time n, before the emission of the post at time n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We also introduce the normalized version of this vector π(n) = {πi(n)}i∈I where the components are the normalized probabilities πi = pi � j∈I pj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Dynamic variables of users (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', their opinion x(u)) and influencers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', their popularity pi) are updated upon every post generation according to Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' It provides a detailed description of the dynamics captured by our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The model’s key features are further illustrated schematically in Figure 2: an influencer posts a message, the social media platform filters it according to ω, and users provide feedback via θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' These two features represent, respectively, an algorithmic effect (function ω): selective exposure, namely the tendency of a platform to suggest similar content to maximize time spent on the social platform, and an individual effect (function θ): confirmation bias, namely the tendency to value content that is close to one’s point of view, as discussed in [1] and the resources therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' These tendencies can explain the appearance of echo chambers in social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' More specifically, in an elementary step of the dynamics, a post is generated by one of the influencers, selected according to the distribution f (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The influencer i posts on its reference direction r(i) = j with a 1In this paper, the verbs “send”, “suggest” and “reach” are used interchangeably referring to a post shown to a user by the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 2From now on, we will refer to regular users simply as users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Moreover, the terms agent or individual are used to indicate a social network user of either class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 6 probability equal to its consistency c(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Otherwise, it posts on one of the other directions in the opinion space j ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', d−1}\\{r(i)} = Nr according to a given distribution, Pr[j = k] for k in Nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In the rest of the paper, we assume for simplicity that this distribution is uniform over the set of non-reference directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We suppose that each post contains exactly the influencer’s opinion on the topic (and that no noise in the user’s perception of the post is present).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Then, note that the post contains a real-valued opinion that is the j-th component of the influencer’s opinion vector x(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In principle, this generated post can reach any user: no explicit network structure is considered for the population3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Following the intuition that individuals with strongly divergent opinions are less likely to interact and therefore, as homophily suggests, like-minded individuals are more likely to interact, the subset of reachable users (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', users to whom the platform sends the post) is determined by considering the opinion distance in the reference direction between each user and the posting influencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Adopting such distance as the central metric influencing the reachable group of users is of utmost importance as it couples the dynamics in different directions, which would otherwise evolve independently of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This posts-users matching process constitutes the content personalization we consider in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Note that the social media platform suggests posts that might interest a user in addition to those that a user explicitly subscribes to (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', follows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Algorithm 1 Description of the Communication Asymmetry model Input: Ni influencers, Nu users, filtering function ω, feedback function θ Output: opinion of each regular user x(u)(n), ∀u Output: popularity of each influencer pi(n), ∀i 1: loop 2: select influencer i according to f (i) 3: select a posting direction j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', j = r(i) with probability c(i), other- wise j is selected uniformly on j ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='.,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' d − 1} \\ {r(i)} 4: pi(n + 1) = pi(n) 5: for all regular user u in the population do 6: x(u) j (n + 1) = x(u) j (n) 7: if Ω � ω(|x(i) ri − x(u) ri |,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' πi(n)) � = 1 then {post proposition} 8: get feedback Θ � θ(|x(u) j − x(i) j |) � 9: if Θ = 1 then {positive feedback} 10: x(u) j (n + 1) = αz(u) j + βx(u) j (n) + (1 − α − β)x(i) j 11: update popularity of i: pi(n + 1) += 1/Nu 12: end if 13: end if 14: end for 15: end loop To decide whether a given user is reached by a post (independently from other users),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' we extract a Bernoulli random variable Ω with parameter ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The user receives the message when Ω(ω) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The parameter ω can be interpreted as a visibility function from the influencer’s perspective, as it affects the subset of users reached by its posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' As already mentioned, ω should be a function of the opinion distance in the reference direction dr(n) = |x(u) r (n) − x(i) r (n)| and the popularity ratio πi of the posting influencer, so that the higher the popularity ratio, the more users an influencer can reach on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Users express their feedback to a post on the platform through a Bernoulli random variable Θ � θ(|x(u) j − x(i) j |) � ∈ {0, 1} 3A complete bipartite graph, where I and U are the two sets of nodes and each link has a weight ω computed at each iteration of the dynamics, might represent the underlying network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 7 whose parameter θ depends on the difference in opinion on the actual direction j of the contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Only posts that receive positive feedback, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', Θ = 1, can influence the user’s opinion, reflecting the tendency to ignore unappreciated content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The social media platform collects feedback from all reached users to update the popularity pi of the posting influencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Specifically, the update rule for the popularity of the posting influencer i reads as follows: pi(n + 1) = pi(n) + ΘT (θ, Upost) Nu (1) ΘT (θ, Upost) = � u∈Upost Θ � θ(|x(u) j (n) − x(i) j (n)|) � (2) where Upost is the subset of users who were made aware of the post by the platform, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', those for whom Ω(ω) takes the value one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The summation in the formula gives the aggregate feedback of all users who saw the post, which is normalized by the size of the population of regular users |U| = Nu to update the popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Note that this normalization is introduced only to avoid excessive growth of influencers’ popularity when the number of users becomes large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' It does not affect the system dynamics, as these depend only on the normalized popularity values πi, which are not affected by the scaling factor 1/Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Influencer post generation according to Platform popularity update .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Ω=1 Θ=1 Ω=1 Ω=1 Θ=0 Θ=1 not reached reached Figure 2: Schematic representation of the model dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The figure highlights the proportions of users who view a particular post Upost, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' those for which the random variable Ω takes the value 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' They react with their feedback Θ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', likes), which depends on the opinion distance between them and the influencer i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Then the platform updates influencer’s i popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The core of the dynamic is represented by the opinion update rule, which dictates how the user’s opinion changes on the direction j of the post depending on the previous opinion x(u)(n), the prejudice z(u) and the opinion x(i) conveyed by the influencer through the post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The following system of equations characterizes the updating rule: x(u) j (n + 1) = � � � αz(u) j + βx(u) j (n) + γx(i) j if Ω (ω(dr, πi)) = 1 , Θ (θ(dj)) = 1 x(u) j (n) otherwise (3) where γ = (1−α−β), being the updating rule a convex combination of x(u)(n), z(u) and x(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Whenever the Bernoulli random variable Ω(ω) ∈ {0, 1} assumes the value 0, the post does not reach the user who keeps 8 its opinion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The individual’s opinion is also not affected by the post when the user receives it but does not appreciate it, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', the feedback variable Θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The actual opinion update is a convex (linear) combination of the user’s prejudice z(u) j , the current user’s opinion x(u) j (n) and the belief delivered by the influencer’s post x(i) j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The distance on the reference direction drives the filtering because we assume the platform is unaware of the specific topic associated with the post just created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Note the joint effect in the model of the distance between the user’s opinion and the influencer’s opinion on the reference direction and the distance along the direction defined by the post’s topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Both contribute to determining the likelihood for the user to provide positive feedback to the message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In most OSNs, there are explicit subscriptions to influencers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', the follow mechanism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Our approach does not consider this type of relationship, as we only account for homophilic contacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Since the number of influencers is considerable in practice, homophily is not the only mechanism driving interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' A regular user does not follow all its homophilic influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' However, nowadays, most social media platforms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', Facebook, Instagram, Twitter) not only offer their users content they explicitly subscribe to but also content that users might like based on their activity on the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This resembles the mechanism we are considering in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In our framework, regular users are passive, as they merely consume content produced by influ- encers: this constitutes a rather simplistic assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' First, users can share the posts they receive, which increases their reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Secondly, users themselves write posts that reflect their opinion, influencing other users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The impact of active users is beyond the scope of this article and will be considered in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Observations from Online Social Networks This section presents data from real-world social networks to motivate some of our modeling choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' For a detailed description of the dataset used, see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' One of the most important features introduced in this paper is the concept of reference direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', the main topic an influencer is interested in and on which they publish most of their posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' While this is a reasonable assumption, this claim needs to be supported by evidence from real social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Moreover, we examine the post-generation process to justify the choice of a Poisson Point Process to describe it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Music Politics Sports Food Pandemic Topics 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='8 Fraction of posts Luca Zaia AC Milan (a) 0 10 20 30 40 lag 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 autocorrelation music Luca Zaia AC Milan (b) Figure 3: (3a) Percentage of labeled posts on each of the considered topics for Luca Zaia, an Italian politician, and AC Milan, an Italian football club.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' (3b) Autocorrelation function on a secondary topic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', music, for both influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The reference direction This section shows that influencers prefer to post about a specific topic rather than discuss multiple ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We have developed a post classifier that flags posts based on their topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We should point out that classifying posts on OSNs into topics is not straightforward, and interpreting the results should be done with caution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' First, the range of possible subjects discussed in a social network is practically countless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' For practical reasons, we will only focus on a subset of five topics: Sports, Politics, Food and Cooking, Music, and Pandemics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' These can be considered popular and general enough to cover a substantial fraction of the influencer discussions on OSN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We took a subset of the influencers in the dataset, namely those with the highest number of classified posts (see Appendix B for details on the classification and filtering process on the data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Note that even if the selected topics can be assumed uncorrelated, they are sometimes discussed jointly in one post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In such cases, it is not always clear which is the main topic of the post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' After classification, we examined the distribution of posts on the topics for each influencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In Figure 3a, we show two example influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In these two cases, the influencers have one topic on which they write most of their posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Luca Zaia, an Italian politician, posts mainly about politics, and AC Milan, a soccer club, discusses sports predominantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This behavior supports the existence of a reference direction for influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Figure 4a shows the distribution of the proportion of posts dealing with the main topic of each influencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Recall that this proportion was called consistency in the jargon of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Most influencers have a clear reference topic on which they write more than half of their posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Figure 4b shows the average per-topic percentage of all influencers in the dataset in descending order, regardless of the specific topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' On average, almost ninety percent of the posts are in the reference direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We discovered that influencers with low consistency values are affected by the presence of news outlets in the considered profiles, for which the lack of a sharp main topic is sensible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 Consistency 0 20 40 60 Number of Influencers (a) main 2nd 3rd 4th 5th Topics 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='8 Fraction of posts (b) Figure 4: (4a) Distribution of the fraction of posts published on the main topic of interest by the subset of influencers considered in this experiment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', their consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' (4b) The average percentage of labeled posts on each topic in decreasing order for all the influencers considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The 95% confidence interval for each average value is reported in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Independence of posts’ generation process on secondary directions The way users interact in an OSN is by posting content (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', text, images, videos) and receiving sugges- tions about what other users of the OSN posted, according to the filtering process set up by the social media platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Since influencers’ posts have a much greater reach than those of regular users, they are the focus of our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Namely, we examine the correlation between posts on each topic by looking at the chronological sequence of the messages of the individual influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Our primary focus now is on secondary topics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', the topics that are not the reference for the influencer, as they post less frequently in these topics, and one might expect to observe a bursty posting behavior, not well captured by the Poisson process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 10 In the previous section, we were able to assign a reference direction r(i) to each influencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Here we look at the time series of the Influencers’ labeled posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' For each secondary direction s(i) j , we define an indicator function 1{postlabel=s(i) j } that takes the value 1 if the post was labelled as s(i) j and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' For each influencer, we thus obtain four sequences (recall that we consider five topics in total) of Bernoulli random variables indicating whether a post belongs to that particular direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' For these sequences, we calculated the autocorrelation function a(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Figure 3b shows two examples of such autocorrelation functions, limited to 40-time lags, for the profiles of Luca Zaia and AC Milan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The time is discretized, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', the actual time between postings is not taken into account: only the posting events matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' An autocorrelation that equals zero everywhere except at τ = 0 would represent uncorrelated samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In our case, the autocorrelation takes moderate values in most cases (≪ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Therefore, it is reasonable to assume that the post-generation is independent, and a Poisson Point Process is an appropriate choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Lastly, note that the autocorrelation function for the pandemic topic takes larger values than for the other topics (see Figure 5), suggesting that the samples are weakly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This fact is due to the exceptional public interest in the topic and because the outbreak of the epidemic only interested the last part of the considered time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='00 sports politics music 0 10 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='00 food 0 10 20 30 40 pandemic 0 10 20 30 40 average lag autocorrelation Figure 5: Mean autocorrelation values of the post-generation process for each secondary topic of all the influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The last plot represents the average value over all topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The 95% confidence interval is shown in each plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Asymptotic Analysis of the Model This section is devoted to the analytical study of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In particular, results are obtained using a mean-field approach, considering Nu → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In this situation, the equilibrium value for the influencers’ mean- popularity ratios ¯πi and users’ mean opinion value ¯x(z) (which depend on prejudice z) can be analytically determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Furthermore, transient analysis of the system can be carried out by describing the dynamics of the users through a Fokker-Plank equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' For simplicity, we restrict our investigation to the situation where the opinion space is one-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' However, we remark that it is possible to extend the analysis to the more general case by following the same approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Mean field approach When the number of users grows large, it is convenient to characterize the system state by the users’ opinion distribution over the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Moreover, hereinafter we will refer to system dynamics over continuous time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Let (X(t), Z(t)) = (X(t), Z) be the current position (opinion) and prejudice of a randomly selected user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We introduce the cumulative distribution function F(x, z, t) = Pr[X(t) < x, Z < z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The corresponding probability density function is f(x, z, t) = ∂2 ∂x∂zF(x, z, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Note that, by hypothesis, there are no dynamics along the z-axes, thus h(z) = � x f(x, z, t)dx does not depend on t and corresponds to the initial distribution of users’ prejudice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2, we will derive a Fokker-Plank equation for the evolution of the opinion distribution over time and space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' For what concerns the evolution of the popularity of a generic influencer i, recall that we distinguish between its absolute popularity value pi(t) and the normalized value πi = pi(t) � j pj(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We can already write down the equation for the evolution of the mean popularity ¯pi(t) (we remark that influencer’s popularities concentrate around their average as Nu grows large,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' as it can be easily shown): d¯pi(t) dt = 1 Nu λf (i) � x � z f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' t) θ � |x − x(i)| � ω � ¯πi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' |x − x(i)| � dz dx (4) Indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' the rate at which the popularity of influencer i grows is proportional to its posting rate (term λf (i)) times the probability that a generic user at (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' z) provides positive feedback to the post generated at time t (integral term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Moreover, recall from Algorithm 1 that each positive feedback increases the absolute popularity of the influencer by 1/Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Fokker-Planck equation for the opinion distribution In this section, we derive a mean-field Fokker-Planck (FP) equation for the population’s opinion distri- bution, assuming that the number of users grows large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' To be specific, in the continuous-time FP approximation, we assume that for the effect of a post, “users/particles” reach their new position by moving at a constant speed during the interval ∆T equal to the average time 1/λ that elapses between the generation of two successive posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Therefore, assuming that at time t a post is generated by user i, the following equation describes how the opinion of a user with prejudice z evolves from t to t + ∆T: x(t + ∆T) = αz + βx(t) + γx(i)(t) Thus, the increment is: ∆x(i) = x(t + ∆T) − x(t) = α(z − x(i)(t)) + (1 − β)(x(i)(t) − x(t)) (5) where we remark that ∆x(i) here represents the change in position of a user in position x, providing positive feedback to a post of influencer i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We can then compute its average velocity as: E[vx(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' t) | X(t) = x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Z = z] = E �[X(t + ∆T) − X(t) | X(t) = x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Z = z] ∆T � = � i λf (i)∆T θ � |x − x(i)| � ω � ¯πi(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' |x − x(i)| � ∆x(i) ∆T = � i λf (i)θ � |x − x(i)| � ω � ¯πi(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' |x − x(i)| � ∆x(i) (6) where θ � |x − x(i)| � is the probability of providing positive feedback (users move only in this case),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' while ω � ¯πi(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' |x − x(i)| � is the probability with which a user in x is exposed to a post created by influencer i at 12 time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Indeed, users only move if they are exposed to the post and provide positive feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Note that, to avoid a cumbersome notation, we have omitted the dependency on the time of the distance term |x − x(i)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The variance of the velocity is given by the relation: σ2 x(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' t) = � i λf (i)∆Tθ � |x − x(i)| � ω � ¯πi(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' |x − x(i)| � (∆x(i) − E[vx(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' t)]∆T)2 (∆T)2 = 1 (∆T)2 � i f (i)θ � |x − x(i)| � ω � ¯πi(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' |x − x(i)| � (∆x(i) − E[vx(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' t)]∆T)2 This allows us to write down a Fokker-Plank equation [38] for the probability density function f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' t) where x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' z ∈ [a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' b]: ∂f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' t) ∂t = −∂vx(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' t)f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' t) ∂x + 1 2 ∂2σ2 x(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' t)f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' t) ∂x2 (7) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Steady state analysis Now we direct our attention to the existence of stationary solutions for the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Stationary solutions of (7) necessarily satisfy: ∂ ∂x � −vx(x, z)f(x, z) + 1 2 ∂σ2 x(x, z)f(x, z) ∂x � = 0 where vx(x, z) and σ2 x(x, z) must be constant over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This requires normalized popularities to be static (i,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' ω(·) to be constant over time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' From previous equation, integrating both sides with respect to x, we get: � −vx(x, z)f(x, z) + 1 2 ∂σ2 x(x, z)f(x, z) ∂x � = c0(z) (8) where c0(z) is a uni-dimensional arbitrary in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Now, observe that, for every z, previous equation is a first order linear ODE in x, and therefore an explicitly solution for f(x, z) can be obtained: f(x, z) = � c1(z) exp(A(x, z) − A(a, z)) + c0(z) exp(−A(x, z)) � x a exp(A(θ, z))dθ � h(z) (9) where A(x, z) = � x a η(u, z)du η(x, z) = −2vx(x, z) − 1 2 ∂σ2 x(x,z) ∂x σ2x(x, z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Function c0(z) can be obtained by imposing boundary conditions: � −vx(x, z)f(x, z) + 1 2 ∂ ∂xσ2 x(x, z)f(x, z) � ��� x=a,b= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' ∀z which leads to c0(z) = 0, while function c1(z) is determined by imposing the normalization condition: � f(x, z)dx = h(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Observe that when σ2 x(x, z) → 0 and ∂σ2 x(x,z) ∂x → 0, from (8), with x0(z) = 0, we obtain that necessarily the mass concentrates around points for which vx(x, z) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Such points, improperly referred to in the following as equilibrium points, will be characterized analytically later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Turning our attention to popularity dynamics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' recall that stationary conditions necessarily imply nor- 13 malized popularities to be constant over time: ¯πi(t) = ¯πi ∀i On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' absolute popularities naturally grow over time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' but the ratio between any two of them (say i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' j) must converge to a constant value cij equal to the ratio of their corresponding normalized popularities: ¯pi(t) ¯pj(t) = cij = ¯πi ¯πj ∀i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' j ∈ I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' i ̸= j (10) Now observe that in stationary conditions the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' of (4) does not depend on time, therefore (4) admits the following trivial solution: ¯pi(t) = � λf (i) � x � z θ(|x − x(i)|)ω(¯πi, |x − x(i)|)dF(x, z) � t Nu + ¯pi(0) (11) Therefore, we meet conditions (10) for any t ≥ 0, iff normalized popularities of influencers {¯πi}i satisfy the following system of equations: λf (i) � x � z θ(|x − x(i)|)ω(¯πi, |x − x(i)|)dF(x, z) = c¯πi ∀i, for some c ∈ R+ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' ¯πi ≥ 0 and � i ¯πi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' (12) and the initial condition {pi(0)}i satisfies (10) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' pi(0) = k¯πi for some k > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Let ki(¯πi) := λf (i) � x � z θ(|x − x(i)|)ω(¯πi, |x − x(i)|)dF(x, z) ¯πi ∈ [0, 1] (13) We can show that: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Solutions of (12) always exist whenever ki(·) ∈ C1[0, 1], ki(·) is increasing, continuous and strictly concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The proof is reported in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We remark that when ki(0) > 0 ∀i, the solution is always unique with ¯πi ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Instead when ki(0) = 0 for some i, the solution is not guaranteed to be unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Now, the problem is how to jointly solve for stationary solutions of {¯πi}i and F(x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In a schematic way, on the one hand, we have shown that given ¯π = {¯πi}i, and h(z), we can uniquely determine a F¯π(x, z) = H(¯π), where F¯π(x, z) = � x −∞ � z −∞ f¯π(y, w) dy dw is the opinion distribution of users resulting from fixed influencers’ popularities ¯π (by (9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' On the other hand, under the conditions: ki(·) ∈ C1[0, 1], ki(·) is increasing and strictly concave, ki(0) > 0 ∀i, given F(x, z), we can obtain a ¯πF = G(F(x, z)) that uniquely corresponds to i (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The existence of a unique fixed point for the joint system of (stationary) users’ opinions and influencers’ popularities is guaranteed under the condition that the operator H ◦ G(·) is a contraction over a complete space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Under the assumption that both ω(·, ·) and θ(·) exhibit a sufficiently weak dependence on their variables, the operator H ◦ G(·) is a contraction over a complete space, and therefore a unique stationary solution exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The proof is reported in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Asymptotic analysis of the fluid limit Previous theoretical analysis is, unfortunately, non-constructive, meaning that it does not allow for direct computation of stationary solutions of our dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' To complement the previous analysis, in this 14 section we propose a methodology to compute numerically stationary solutions, even in multi-dimensional scenarios, under the assumption that Nu → ∞, β → 1, σ2 x(x, z) → 0 and ∂σ2 x(x,z) ∂x → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In the following, we will refer to the such regime as fluid limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Mean opinion assuming that normalized popularities converge As already observed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='3, recall that, given ¯π = {¯πi}i, the distribution of users with a given prejudice z concentrates around equilibrium points, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', points ¯x(z) at which v(x, z), as given in (6), is null (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' v(¯x(z), z) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' points ¯x(z) must satisfy equation: 0 = � i f (i)ω � ¯πi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' |¯x − x(i)| � θ � |¯x − x(i)| � � α(z − x(i)) + (1 − β)(x(i) − ¯x) � (14) Defining for compactness di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='¯x = ��¯x − x(i)�� and recalling γ = 1 − α − β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' from (14) we get: ¯x(z) = α 1 − β z + γ 1 − β � i∈I f (i)ω � ¯πi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='¯x� θ � di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='¯x� x(i) � i∈I f (i)ω (¯πi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='¯x) θ (di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='¯x) (15) The assumption β → 1 is required to avoid too large oscillations of users’ opinions in response to a single post generated by an influencer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' which may reduce the accuracy of our mean-field approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This hypothesis is not restrictive: since β represents the weight individuals give to their current opinion, we can reasonably assume that users do not dramatically change their opinion in response to single post events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Normalized popularities assuming opinion convergence Here we assume that users with prejudice z are concentrated in opinion point ¯x(z), and we look for the stationary popularity ratios ¯πi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' To simplify the expressions, we introduce the quantity Fi(¯πi) ≜ � z f (i)ω(¯πi, di,¯x(z))θ(di,¯x(z))h(z)dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Observe that solutions of (12) are necessarily in the form: ¯πi = Fi(¯πi) � j∈I Fj(¯πj) (16) where c appearing in (12) is given by c = 1 � j∈I Fj(¯πj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Under the assumption that ω(·, ·) is concave in its first argument (for any choice of the second), Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1, guarantees the existence of such solutions for every choice of function ¯x(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Moreover, even in the more general case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', when ω(·, ·) is non-concave in its first argument, solutions of (16) can be found numerically in many cases, through a fixed point iteration method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' To conclude, observe that a pair (¯x(z), {¯πi}i) represents a stationary solution if it jointly satisfies (15) and (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The existence of such solution can be, again, only verified numerically through a fixed point approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' At last, note that, in the special case in which all users have the same prejudice z we can rewrite (15) as: ¯x = α 1 − β z + γ 1 − β � i∈I Fi(¯πi, ¯x)x(i) � i∈I Fi(¯πi, ¯x) = α 1 − β z + γ 1 − β � i∈I ¯πix(i) (17) which provides a direct formula for the mean opinion ¯x in terms of the normalized popularities ¯πi and the influencers’ opinions x(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Model predictions In this section, we present a selection of results obtained while varying the model parameters, providing valuable insights into the impact of algorithmic personalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Results are obtained through a Monte Carlo 15 Table 1: Parameters and functions shared across experiments Symbol Value - Form Description Ni 2 Number of influencers x(0) j 0 Opinion of influencer 0 on direction j x(1) j 1 Opinion of influencer 1 on direction j r(0) 0 Reference direction of influencer 0 r(1) 1 Reference direction of influencer 1 p0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1(0) 100 Initial absolute popularity of both influencers Nu 10000 Number of regular users Niter 100000 Number of iterations for each simulation α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='05 First weight in the updating rule in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 3 β 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='93 Second weight in the updating rule in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 3 θ(·) 1 − ���x(i) j − x(u) j ��� Functional form of the feedback function ω(·) e−ρ(x(u) r −x(i) r ) 2 πi Functional form of the visibility function 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='00 x0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 x1 Figure 6: Table 1 summarizes some of the parameters of the system shared across the different experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' On the right-hand side, a realization of the initial opinion distribution of the regular users (being also the prejudice since z(u) = x(u)(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We focus on the two main dynamic variables of the system: the average opinion ¯x of regular users and the normalized popularities {¯πi}i for influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Note that the quantities shown in this section, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', the pair (¯x, ¯π), are obtained as empirical averages over multiple runs, and over all the regular users, as far as ¯x is concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Hence, they are different from the values presented in the previous section, in principle, which pertains to the limiting case of an infinite population of users with the same prejudice z, and where β approaches 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Moreover, in some cases, to save space, we omit the results on average user opinion because it is tightly coupled with the normalized popularities, as observed in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Lastly, to facilitate the interpretation of results, we restrict ourselves to the case of two “competing” influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We provide further details on the scenario considered in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The model can clearly be applied to scenarios with an arbitrary number of influencers occupying any position in the opinion space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2, we present the behavior as function of publication frequency f (i), and in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='3 as function of consistency c(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Then we show examples of final opinion distributions of the regular users in a few paradigmatic cases in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Finally, in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5, we consider the degree of stubbornness, defined as δ = α γ , which governs the opinion update of the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Description of the scenario The default parameters of our reference scenario are reported in Table 1 unless otherwise explicitly stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' As mentioned earlier, we restrict ourselves to the case of two “competing” influencers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', Ni = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We assume that x(0) j = 0 and x(1) j = 1 ∀j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We consider the case of different reference directions r(0) ̸= r(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In this section, we consider a two-dimensional opinion space and assume that the vast majority of the regular users’ population initially takes a moderate position on both topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' More precisely, initial opinions are distributed according to a Beta distribution, independently on each axis, with shape parameters a = b = 10, as shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Recall that we assume for simplicity that the prejudice of the user z(u) corresponds to the initial opinion x(u), hence Figure 6 also provides the prejudice distribution of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The functional form of visibility ω and feedback θ is also reported in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We take as ω(·) a Gaussian function similar to the trust function in [39], but modulated by ¯πi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Here, the coefficient ρ is a parameter that controls the extent to which the social media platform filters content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Small values of ρ correspond to smooth personalization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', influencers can reach users whose opinion strongly differs from theirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Conversely, high values of ρ correspond to sharp personalization: only close users (in the opinion space) are reachable with non-negligible probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The function θ(·) is assumed to be a decreasing, linear function of the opinion difference dj = |x(u) j − x(i) j |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 16 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Behaviour as function of the frequency of publication The frequency of publication f (i) is one of the basic parameters that characterize influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The higher f (i), the higher the structural advantage of the influencer because it more frequently reaches users through posts, attracting them to its own opinion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In this section, we examine the value of mean normalized popularity ¯π0 as a function of f (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Note that in the case of two influencers, f (1) = 1 − f (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We performed this experiment by fixing the consistency of the two influencers: c(0) = c(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='8, which is approximately the average consistency observed on real-world data (Figure 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5 f(0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5 0 Values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 Figure 7: Popularity ratio ¯π0 of influencer 0 as function of the publication rate f(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Each point is obtained by averaging over 100 time samples and 10 different process’ realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Different levels of personalization are considered by varying the parameter ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The two influencers have the same consistency c(0) = c(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Note that, in the considered scenario, the curves are symmetric for values of f(0) in [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5 f(0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='65 Distance from i = 0 on r(0) Values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5 f(0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='65 Distance from i = 1 on r(1) Values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 (b) Figure 8: Opinion distance of the influencers’ opinion on their reference direction x(i) r(i) and the average opinion of the regular users’ population on the same direction ¯xr(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Various degrees of personalization are considered, tuning the parameter ρ, the setting is the same as that of Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In Figure 7, we consider different levels of personalization by varying the parameter ρ in the exponent of the visibility function ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We see that the higher the degree of personalization (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', the higher the value of ρ), the lower the normalized popularity of influencer i = 0, for any given f (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This result suggests that algorithmic personalization favors the structurally advantaged individual, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', the one with higher f (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This mechanism, in turn, leads to more radical positions in the population of regular users, as the platform preferentially exposes them to the belief of the advantaged influencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Figure 8 clearly shows this behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 17 Note that for high values of ρ, the average user opinion exhibits a significant bias toward the structurally advantaged influencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Such bias persists up to a critical value of posting frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' For example, when the personalization parameter is ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0, the critical posting frequency value is roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='35;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' when the personalization parameter is ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5, the critical posting frequency value is approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We argue that content filtering in OSN poses a potential threat to opinion diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This premise is inextricably linked to the goal of usage maximization [32] pursued by the social media platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Many platforms indeed prefer to suggest just similar content rather than exposing individuals to radically different opinions, allowing for so-called serendipity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Behaviour as a function of the consistency In section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1, we showed the existence of a reference direction for real influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Here, we investigate the impact on dynamics of the extent to which an influencer publishes on its reference direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', its consistency c(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In this experiment, we consider two influencers with the same posting frequency f (0) = f (1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5, and we let c(0) vary while keeping c(1) fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We then consider different choices of c(1) to grasp its impact on the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' From Figure 9, we see that consistency does not significantly affect the normalized popularities when personalization is smooth (ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0001), while it becomes relevant when the platform applies sharp personalization to the content (ρ = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 c(0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='7 0 Values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 (a) c(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 c(0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='7 0 Values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 (b) c(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 c(0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='7 0 Values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 (c) c(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 Figure 9: Popularity ratio ¯π0 of i0 as function of its consistency c(0), while considering f(0) = f(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5 and keeping fixed the consistency of the second influencer at (9a) c(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5, (9b) c(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='8 and (9c) c(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The two colors represent two different levels of personalization (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', smooth and sharp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Each point is obtained by averaging over 100 time samples and 10 different realizations of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' It is interesting to observe that the maximum of ¯π0 moves to the left (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', is achieved for a lower value of consistency c(0)) as the consistency of i = 1 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 18 Before discussing the results concerning sharp personalization shown in Figure 9, it is crucial to note that when considering a bi-dimensional opinion space, a consistency c(i) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5 implies that the choice of the reference direction for influencer i is somehow unnatural since it produces the majority of posts on the other direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This choice is in contradiction with the definition of reference direction itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Nonetheless, we leave this situation as a possibility: let us imagine an influencer can adopt consistency values of less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5 while undertaking a transition phase during which it changes its main topic for its posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In this case, the platform would still perform personalization on the given reference direction r(i), but the consistency would be less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5 due to the change in posting pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' It represents a scenario of interest, and as such, we allow for c(i) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' First, we note that the shape of the curves in Figure 9 depends strongly on the value of the consistency of the “opposing” influencer c(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Second, a perfectly balanced condition is achieved whenever the two influencers have the same consistency since all parameters are symmetric (even the curves associated with ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0001 and ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 coincide on this point), see Figure 9c at c(0) = 1 for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The observed pattern is consistent in all three figures with ¯π0 being first increasing and then decreasing, exhibiting a unique maximum in all three diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Let us start the discussion by considering Figure 9c because its interpretation is instrumental to better understanding the other scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' It represents a rather degenerate situation since the influencer i = 1 posts exclusively in its reference direction r(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' However, the simplicity of the scenarios allows us to interpret the results straightforwardly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In this case, the influencer i = 0 has r(0) = 0 and for 0 < c(0) < 1 4 posts in both directions, with the social media platform filtering according to distance in the reference direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In the direction r(0) = 0, the influencer has no competition at all, since c(1) = 1, so it is able to attract the user population to its “reference opinion” while competing with the other influencer in the non-reference direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The lower the consistency c(0), the greater the competition on r(1) = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', for values of c(0) close to 0, the influencer i = 0 posts the vast majority of its messages on r(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Therefore, the final value of ¯π0 reaches higher values, as demonstrated in Figure 9c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' It happens because the influencer i = 0 has a stable feedback stream from the posts in its reference direction, where it does not face any competition, and it competes with i = 1 in the other direction, being at an advantage since the visibility of its posts is determined by the opinion distance in its reference direction r(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' From these observations, we can conclude that influencer i = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', the influencer with the higher consistency, is disadvantaged for virtually all values of the other influencer’s consistency c(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Here we are considering the extreme case, where the influencer i = 1 has the maximum attainable consistency c(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Thus, we can easily conclude that the individual with lower consistency holds the structural advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This is consistent with the results in Figure 9a and 9b, for which we observe a decrease in normalized popularity ¯π0 on the right of the point at which c(0) = c(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Namely, the influencer with higher consistency is penalized and eventually reaches lower values of ¯πi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This assertion is not true in the first part of all plots in Figure 9 where ¯π0 is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', the influencer i = 1 is favored, even though c(0) < c(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' It is important to note that this is only true for values of c(i) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5, which, as discussed above, are only relevant in certain situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This behavior depends on the interplay between the actual main posting direction (which is different from the reference direction when c(i) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5) and the algorithmic personalization performed on r(i), see also the discussion in the footnote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Interestingly, the curves intersect for the first time when c(0) ≈ 1 − c(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' From our results, choosing a consistency c(i) around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5 for i = 1 seems to be a successful choice: from Figure 9a it is clear that the influencer i = 0 can only reach and never exceed the value of the normalized popularity of its “opposing” influencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In Figure 9b we assign a higher consistency c(1) for the competitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The discussion developed above applies, and for values of c(0) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5, the influencer with the lower consistency always has an advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Indeed, we find that for consistency values c(0) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='8 = c(1) (and c(0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2) influencer i = 0 achieves higher values of ¯πi, see Figure 9b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In summary, the results of this section suggest that a given influencer can gain an advantage over its 4The point c(0) = 0 corresponds to a rather peculiar situation where the influencer i = 0 does not post on its reference direction, but only on the other direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Then the filtering depends on the initial configuration of the users in this direction, where no dynamics occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' It can be concluded that the scenario is fairly balanced, with the influencer i = 1 having a slight advantage since the filtering occurs in the direction where the dynamics take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 19 competitors if it has a lower consistency c(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This observation also reflects the natural tendency of people to seek varied content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Opinion configuration considering combinations of reference directions In previous sections, we have focused primarily on the influencer perspective, looking at normalized popularity values ¯πi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Here we present possible final opinion configurations in scenarios in which the reference directions of influencers (r(0), r(1)) are either coincident or different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' It was observed above that when an influencer has a structural advantage, it achieves a higher πi and, in turn, can exert a higher attracting force to the regular users towards its opinion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We then argue that it is interesting to examine what can happen in a symmetric scenario in terms of frequency of publication (f (0) = f (1)) and consistency (c(0) = c(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' As before, the influencers hold opinions x(0) = (0, 0) and x(1) = (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We consider the two possible cases where the two opinion leaders have either the same or different reference directions and the impact of algorithmic personalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This symmetrical scenario is interesting because, in most cases, it guarantees the coexistence of both influencers (neither of them ‘wins’), see the normalized popularity plots in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Therefore, the final opinion distribution is the result of the joint influence of both agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In Figure 10a, we observe only a negligible perturbation with respect to the initial distribution shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In this case, the platform practically does not filter the content, so every post reaches all users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' From a regular user perspective, individuals are exposed to nearly identical forces, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', “opposite” stimuli from the two influencers, which almost perfectly cancel each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In Figure 10b, the impact of strong personalization is clear: the filtering effect introduced by the platform leads to the emergence of two echo chambers, whose membership is determined mainly by user’s prejudice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Each user reaches an equilibrium point at which the resultant attraction induced by the two influencers is balanced by the attraction exerted by its own prejudice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Interestingly, users also tend to cluster in the non-reference direction (x1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 10b) and align their opinion with that of the influencer associated with the echo chamber they end up in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We remark that this is a metastable condition, as the πi diagram indicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' By extending the time horizon, we may observe a different final situation in which one of the two influencers “wins” (exhibiting behavior similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 10d, but just taking place at a different time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=') Figures 10c and 10d refer to the case of different reference directions: the two influencers do not compete on the same topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In Figure 10c, it is clear that there is no competition as the two influencers are able to attract users to their reference opinion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', x0 = 0 the reference opinion of i = 0 and x1 = 0 that of i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' It constitutes a particularly relevant case, whose occurrence is linked indissolubly to the newly introduced concept of reference direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In the last scenario, shown in Figure 10d, the influencer i = 1 “wins ”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', ¯π1 → 1, which brings public opinion closer to their belief on both issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The users’ opinion does not overlap with that of the winning influencer because they are anchored by their prejudice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In this case, an unstable behavior is observed since the identity of the winner influencer (as expected, as a result of perfect symmetry) depends on random factors, and different sample-paths lead to diverse winners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' It should also be noted that sharp personalization leads to a situation where the public scene is monopolized by only one individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Behaviour as function of the updating weights The behavior of the system depends not only on the characteristics of the influencers and the composition of public opinion, but also on the parameters controlling the opinion update rule in equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The update is a convex combination of the prejudice, the current opinion, and the opinion conveyed by the post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We chose to hold fixed the weight β associated with the current opinion and consider the ratio of the other two weights α γ , which we termed degree of stubbornness, as it gives an indication of the extent to which users change their opinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We considered an unbalanced scenario in which influencer i = 1 has a structural advantage, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', f (1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='7 > f (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Figure 11 again shows that personalization favors the structurally advantaged indi- vidual (consistent with section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Note that the x-scale is logarithmic to highlight the sudden drop of ¯π0 for α γ ≈ 10−3 (corresponding to modest values of α) when sharp personalization is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The shape of the two curves is quite similar, only the decrease is observed at different values of α γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Smooth personalization 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='00 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 x1 0 20000 40000 60000 80000 100000 n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 i 0 1 (d) Figure 10: (10a,10b) Influencers with the same reference direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', r(0) = r(1) = 0, while (10c,10d) influencers have r(0) = 0 and r(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In both cases, two different degrees of personalization are considered: smooth (left column, ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0001) and sharp (right column, ρ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In all cases, the influencers have consistency c(0) = c(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The distributions were obtained as the time average of the opinion distribution in one realization of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The normalized popularities of the two influencers in the given realization are shown along with the distributions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' it is clear that the influencers coexist except in 10d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 21 allows the coexistence of influencers on the whole domain, while with sharp personalization, for a wide range of parameters, influencer i = 1 “wins.” In both cases, there is an initial phase (for low values of α γ ) in which the two influencers coexist, and this is followed by a drop of the normalized popularity of the disadvantaged influencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This can be explained by the fact that small values of α γ imply that a negligible weight is given to the prejudice, and therefore regular users concentrate around the two influencers’ opinions on their reference direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This can be easily confirmed by looking at the final opinion configuration of users, who concentrate in the upper corners of the opinion space (around [0, 1] and [1, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This is because the influencer i = 1, whose opinion is x(1) = [1, 1] is stronger than the other in terms of popularity and is able to pull users along its non-reference direction as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We remark that when users are very close in opinion to a particular influencer, it is difficult for the other to persuade them, as the probability of this happening is proportional to the product ω · θ, both of which are a function of opinion distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In these scenarios, the distance from the “further ” influencer is dj ≈ 1, which drastically reduces the probability of reaching the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Thus, as long as α γ is small enough, both influencers can build their user base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' These situations represent rather degenerate cases where the population almost disregards their prejudice in favor of the opinion conveyed by the post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' It might be interesting to consider users with varying degrees of “volatility” who are able to pull along the opinion of their neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' As the degree of stubborness increases, so does the inertia of the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' They are more entrenched in their prejudice and therefore no longer concentrate in a small neighborhood of the influencer’s opinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This favors the structurally advantaged influencer, as the other (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', i = 0) is unable to build its user base because users do not get close enough to it (see Figure 11 for i = 1 and ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0001 we have ¯π0 → 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The subsequent rise in ¯π0 depends on the fact that when α approaches the maximum value αmax = 1 − β, users give importance only to their prejudice, and therefore they do not deviate too much from their initial position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' As a consequence, it can not be triggered the positive feedback between users’ opinion and influencers’ popularity that leads to the complete victory of one influencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 10 4 10 2 10 0 degree of stubborness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='30 0 Values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 Figure 11: Normalized popularity ¯π0 as a function of the degree of stubbornness α γ , the points are obtained considering 50 realizations of the process and averaging over 100 discrete time instants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Again, two levels of algorithmic personalization are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Analysis of the fluid limit In this section, we compare predictions of the simplified fluid limit against simulation results of the full stochastic model described by algorithm 1 (obtained through a Monte-Carlo approach) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We restrict ourselves to a one-dimensional opinion space, as in section 5, and assume that all users share the same prejudice z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Again, we consider two “competing” influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' A similar analysis could be performed in scenarios with any number of influencers at any point in the opinion space, but this would be computationally more challenging since multiple stationary points may exist, each with its own attraction basin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 22 First, we derive in section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1 some preliminary analytical results for the case of two influencers, using the results of the fluid limit introduced in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Then in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2 two extreme instances of the model are solved in closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='3 is devoted to comparing the analytical results of the fluid model with simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Finally, we discuss the impact of content personalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Two competing influencers Let us specialize the equations presented in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='4 for the mean opinion ¯x(z) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' (17)) and the normalized popularities ¯πi (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' (16)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Note that for Ni = 2, ¯π0 = 1 − ¯π1, so it is sufficient to study ¯π1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' As for the mean user opinion ¯x(z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' equation (17) allows us to write the asymptotic mean directly as a function of ¯π1 and the opinions of the two influencers x(0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' x(1): ¯x(z) = α 1 − β z + γ 1 − β � (1 − ¯π1) x(0) + ¯π1x(1)� (18) Substituting the functional forms of the visibility ω and feedback θ into equation (16),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' we obtain the following expression for the normalized popularity ¯π1: ¯π1 = f (1)e−ρ(x(1)−¯x) 2 ¯π1 � 1 − |x(1) − ¯x| � � i∈{0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1} f (i)e−ρ (xi−¯x)2 ¯π1 � 1 − |x(i) − ¯x| � = f(¯π1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' ¯x) (19) Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' if we combine the above expression with equation (18) for ¯x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' we get ¯π1 = f(¯π1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' which can be solved numerically through a fixed-point approximation (FPA) (a graphical representation is shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The outcome of this FPA and the corresponding simulation results are compared in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Closed form computations in extremal cases The combination of equations (18) and (19) cannot be solved in closed form in the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' However, there are at least two scenarios in which this is possible, separately considered in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' When an influencer “wins” We consider an influencer a “winner” if its normalized popularity ¯πi approaches 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Suppose that the influencer whose opinion is x(1) = 1 wins, then ¯π1 → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This implies ¯π0 → 0 and thus ω → 0+: the influencer with x(0) = 0 is seen by a negligible fraction of users and in practice, only influencer i = 1 remains visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Note that in the extreme case in which influencer 1 wins, users see only x(1), and asymptotically all users move towards it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In this case, the final opinion ¯x(z) can be easily calculated with a recursion of the update rule (3): x(u)(n) = n � i=0 βi � αz + γx(1)� + βnx(0) For n → ∞ and considering β < 1 (the case β = 1 coincides with the trivial case where users remain fixed at their initial opinion) we get: x(w) = α 1 − β z + γ 1 − β x(1), (20) which is in agreement with (18) if one sets ¯π1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This corresponds to one of the extreme cases that we will use later to examine the model behavior as a function of the personalization parameter ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' It should be noted that this construction relies on the knowledge of the winning influencer, which is unknown in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' However in the fluid limit, we expect that the winning influencer, if any, is the one that has a structural advantage over the others at the beginning (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', a higher posting rate f (i), see Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 23 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Constant personalization function The other extreme case we consider is the one in which ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In this case, the personalization function ω no longer depends on ¯πi, and it is easy to see from Table 6a that it returns ω ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Moreover, we consider x(1) = 1, x(0) = 0, which further simplifies (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The above formulas (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 19 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 18) can then be solved in closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In particular, equation (19) for the normalized popularity ¯π1 becomes: ¯π1 = f (1) (q + m ¯π1) f (0) (1 − (q + m ¯π1)) + f (1) (q + m ¯π1) where m ≜ γ 1−β and q ≜ α 1−β z for compactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This leads to a second order equation which can be easily solved for ¯π1: ¯π2 1 m(f (1) − f (0)) + ¯π1 � f (0)(1 − q) + f (1)(q − m) � − f (1)q = 0 (21) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Comparison between analytical prediction and Monte Carlo simulations This section is devoted to comparing the analytical results derived in section 5 with simulations of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Numerical and graphical solutions of equation (19) are also provided, shedding light on the impact of the algorithmic personalization performed by the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Description of the scenario The scenario setting is analogous to that described in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1 and Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='However, here, we consider a one-dimensional opinion space [0, 1] and we assume all users to have the same prejudice, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', z(u) = z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='4, ∀u ∈ U matching their initial opinion x(u)(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The “competing ” influencers have opinions at the extremes of the domain, and their posting frequencies are f (1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='7 and f (0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', influencer i = 1 has a structural advantage over influencer i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Note that in a one-dimensional space, the reference direction r(i), and hence the consistency c(i), lose their significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' To avoid obtaining trivial results in which influencer 1 obviously wins, regular users are initially placed closer to the disadvantaged influencer i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Simulation, fluid limit and fixed-point approximation Comprehensive validation and comparison of the approaches used to obtain the system equilibria are shown in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' First, the stochastic model described by Algorithm 1 is “simulated” by obtaining 100 different sample whose length is 500000 elementary steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The variables of interest ¯x(z) and ¯π1 are obtained by averaging the process over both discrete times steps n and sample paths and are represented by circle marks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Second, equation (18), which is a specialization of (17) obtained from the fluid limit, indicates that the state of the system lies on a line in the plane ¯π1,¯x (dashed line in Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Third, the extreme cases of the model analyzed in section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2, for which we derived a closed-form solution, are represented by star-like marks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Lastly, diamonds are solutions of (19) employing the fixed-point approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We observe that, for given ρ, simulation marks match well with analytical marks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The only exception is for ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5, for which simulations provide ¯π1 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='79, whereas the analysis provides ¯π1 ≈ 1 (see also the table on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This mismatch is due to the fact that ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5 is close to a ‘phase transition’, at which the system switches from a regime in which two stable solutions exist (in particular, one in which both influencers survive) to a regime in which influencer i = 1 wins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This behavior is better illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 13, where the curve corresponding to ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5 is almost tangent to the bisector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' It should be noted that the “empty” region in Figure 12 is directly related to this behavior since no stable solutions can exist for that values of ¯πi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In fact, there is no stable intersection with the bisector in Figure 13 in the corresponding interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Implications of algorithmic personalization We summarize here the insights into algorithmic personalization suggested by the emergent behavior of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We already mentioned how content filtering favors influencers with a structural advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='00 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='56 x simulation fixed-point 1 z + 1 1 closed-form = 0 closed-form 1 1 10 3 10 2 10 1 10 0 in simulation Figure 12: Comparison between analytical results, including the exact extreme points calculated in 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1, and the linear relationship between ¯x and ¯π1 according to Equation (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Diamonds represent the fixed-point approximation for the solution of equation (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Simulation results of the stochastic dynamics, represented by circles, were obtained by averaging 100 realizations of the process as described by Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We consider a scenario in which α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='05, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='93, with two influencers at the extremes of the domain, with f(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='3, f(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='7 and the same initial absolute popularity p0 = p1 = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Numerical values from simulation and fixed-point approximation are reported in the table alongside the plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Table 2: Simulation and FPA ρ ¯π1 ¯π1 SIM FPA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='682 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='684 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 f( 1) = 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='001 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='01 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='8 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 1 Figure 13: Graphical solution of ¯πi = f(¯πi), (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Stable solutions corresponds to intercepts between f(¯πi) and the bisector, such that f′(¯πi) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We observe that non-trivial solutions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', solutions in which both influencers survive) exist, roughly in the interval [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='8], provided that ρ is not too large (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', ρ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' For ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5, the only stable solution is ¯π = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This explains the results in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Simulation results reported on the alongside table confirm the validity of the analytical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' For instance, in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2, we showed that personalization promotes the influencer with higher posting frequency, and, in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='3, the one with lower consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In addition, in 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2, we presented a case 25 where a ‘phase transition’ is observed as a function of the filtering strength (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Indeed, after a certain threshold, the favored influencer (say influencer 1) is the only one that survives (¯π0 → 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In such a situation, the population is exposed to the opinions of a single individual, hindering diversity on the social platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' It is also interesting to discuss the results in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='4 concerning the effects of personalization on the opinion distribution of regular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The two possible outcomes when sharp personalization is applied are: either the emergence of two echo chambers with users holding more radical positions in both directions or the onset of an unstable situation in which the two influencers coexist for a limited amount of time, after which ¯πi → 1 for an i dependent on the specific sample path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Online social network data This section examines data collected from Facebook and Instagram social networks and compares the observed behavior with some of the findings of our Communication Asymmetry model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Correlation between frequency of publication and popularity In previous sections, especially in 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='3, we discussed structural advantage from the influencer’s point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' One of the key advantage parameters, as observed across all experiments, is the publication frequency f (i): the higher f (i), the greater the advantage (see Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In this section, we attempt to validate this finding by correlating the frequency of publication of influencers with their popularity growth, using the total number of followers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', the number of people subscribed to the profile, as a proxy for popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We consider temporal sequences from Instagram on a sample set of 110 influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5 Pearson correlation coefficient 0 5 10 15 20 25 Number of influencers Figure 14: Distribution of the correlation coefficient between monthly number of posts and popularity growth (in terms of number of followers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' For each influencer, we considered a temporal granularity of one month, determined the number of posts during this period, and calculated the relative change in the number of followers considering the values at the beginning and end of the interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Then for each user, we calculated the Pearson correlation coefficient between the number of posts and the relative variation of followers in the month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In Figure 14, we show the distribution of these correlation coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Results suggest that there exists, in general, a positive correlation between the two quantities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', influencers with aggressive posting habits tend (but not always) to get more followers, which likely favors them when in competition with other influencers on social media platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This is consistent with the model predictions shown in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Case Study: Italian government crisis in August 2019 In June 2018, a few months after the general elections, Giuseppe Conte was appointed Italian Prime Minister.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Two parties formed his supporting coalition: Movimento 5 Stelle (his own party, holding the 26 relative majority of the Italian Parliament) and Lega, whose leader was Matteo Salvini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In August 2019, Salvini decided to withdraw Lega’s support to the government, starting a crisis aimed at driving Italians to new elections and gaining more votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' However, Movimento 5 Stelle managed to reach an agreement with various parties to form a new government, and on September 5, 2019, Giuseppe Conte became Prime Minister for the second time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The coalition that supported this new administration clearly excluded Lega.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In this section, we apply the proposed model to reproduce the sudden rise of Giuseppe Conte’s popularity in social networks during the government crisis in August 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We exploit the multidimensional capability of the model considering two directions: Politics, reference topic for Salvini and Conte, and attitude toward government fall (End government, see Figure 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' (a) Initial distribution (b) After the transient Figure 15: Initial distribution density of the population along the Politics direction and End government direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The opinion position of the two leaders in the space is depicted with a green (Salvini) and a yellow (Conte) point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In the opinion space, we assume Salvini has a more radical political viewpoint, while Conte has an opposing and more moderate position (described somehow arbitrarily by putting Salvini at xP olitics = 0, Conte at xP olitics = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='76).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We assume that the population has a moderate initial opinion (centered at xP olitics = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5, see Figure 15a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Regarding the attitude toward government fall, the two politicians obviously have a completely different opinion (Salvini has xEndgovernment = 1 while Conte has xEndgovernment = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We assume that the population is strongly polarized towards Conte’s opinion in this latter direction (Figure 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This is an a posteriori assumption made knowing the outcome of the social confrontation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' A period of eleven weeks is considered, from July 7 to September 22, during which data was collected weekly from Facebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' A total of 1162 posts were published, of which 125 were by Conte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The rate f (i) is calculated as the number of posts by an influencer relative to the total number of posts (f (Conte) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='108, f (Salvini) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='892).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Some simplifying assumptions are necessary to apply the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We assume that the two politicians have a consistency c(i) of exactly one (real values are often close to this value, see Figure 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Moreover, Giuseppe Conte and Matteo Salvini are the only influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Although this hypothesis is restrictive, in the scenario studied, the two influencers were the main (active and popular) protagonists during the government crisis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Moreover, we consider the simplest scenario in which personalization is not employed: ρ = 0 and thus ω ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We consider a feedback function of the form θ = e−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='25(x(u)−x(i))2 for both opinion directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' For an exhaustive list of the parameters, we refer the reader to Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 27 transient phase 7th July 22nd Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' switch posts along politics government crisis 28th July 1st Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' switch posts along politics posts along end government 0 Figure 16: Timeline of modelled scenario from July 7, to September 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' From July 28 to September 1 we have a consistency switch, with posts along End government direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Figure 16 shows the timeline of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The two influencers start with the same initial popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We consider a transient of Nt = 10000 discrete time-units, after which the stationary normalized popularities πi roughly correspond to the empirical normalized popularities obtained by dividing the number of followers of each influencer by the total number of the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' After the transient, we can see in Figure 15b that the distribution of public opinion is skewed towards Salvini, who, in turn, has a higher popularity ratio due to his higher publication frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' After the transient, the crisis happens and both influencers start posting in the Endgoverment direction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', we observe a consistency shift for both influencers), during a time window of five weeks that approximates the duration of the government crisis, after which the two politicians switch back to posting on the Politics direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Note that the initial users’ opinion distribution along the Endgoverment axes is concentrated around Conte’s point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Even with these limitations, it is still possible to reproduce the observed social behavior as a whole: it corresponds to a situation where an influencer is in stark contrast to the opinions of its user base and loses ground with respect to the other influencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Note that, in the model, only a very unbalanced distribution of the population towards Conte’s opinion (against the government fall) can explain the sudden increase in Conte’s popularity, despite the remarkable differences in popularity ratios in favor of Salvini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Figure 17 compares the simulation results of the described setting and Facebook’s measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Note how the model can explain the sudden rise in Giuseppe Conte’s popularity, precisely in the weeks of the government crisis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Clearly, our model does not exactly fit empirical observations but simply provides qualitative insights into the possible causes of the rather sudden popularity shift that was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Many of the model’s parameters are unknown, such as the opinion distribution, the weights of the updating rule, or the feedback function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' However, by making reasonable assumptions about some of the parameters, one can obtain a reasonably good fit, and exploit the explanatory capability of the model to acquire better confidence in the hidden mechanisms beneath observed dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In this experiment, we followed exactly this approach and we looked for some mechanisms that could justify the same sudden surge in popularity that occurred during the government crisis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' As the main outcome of our analysis, we conclude that the observed popularity trends of the two considered influencers can be largely explained by considering the fear of political instability in the user base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Conclusions In recent times, online social interactions appear essential to human relationships and play an increasingly important role in opinion formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' To understand the mechanisms underlying this novel communication paradigm, it is of utmost importance to develop flexible frameworks suitable for describing interactions on social media platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In this work, we have developed an opinion model tailored to online interactions, with 28 07 Jul 14 Jul 21 Jul 28 Jul 04 Aug 11 Aug 18 Aug 25 Aug 01 Sep 08 Sep 15 Sep 22 Sep 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='220 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='230 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='235 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='240 Conte Model outcome Facebook data Figure 17: The popularity ratio πConte for Conte, the one obtained from Facebbok data and the one from the model along with its 95% confidence interval, computed over 10 realizations of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' It can be seen how the model follows the increase in popularity during August 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Table 3: Parameters and functions for the Case Study Symbol Value - Form Description Ni 2 Number of influencers x(Conte) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='76 Opinion of Giuseppe Conte on direction j x(Salvini) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 Opinion of influencer 1 on direction j f(Conte) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='108 Opinion of Giuseppe Conte on direction j f(Salvini) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='892 Opinion of influencer 1 on direction j r(Conte),(Salvini) 0 Refrence direction of both influencers pConte,Salvini(0) 20 Initial absolute popularity of both influencers Nu 10000 Number of regular users Niter 15000 Number of iterations for each simulation Nt 10000 Duration of the transient phase w 550 Length of the government crisis α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='3 First weight in the updating rule in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 3 β 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='65 Second weight in the updating rule in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 3 θ(·) e−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='25(x(u)−x(i))2 Functional form of the feedback function ω(·) ρ = 0 =⇒ ω ≡ 1 Functional form of the visibility function particular attention to distinguishing between two classes of users, namely regular users and influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We characterized the influencers by introducing the concept of reference direction, which links unrelated topics discussed by the same influencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Measurements collected from real online social networks support our modeling assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Similarly to other works in the recent literature, we integrated algorithmic personalization in a flexible and tunable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We have shown how content filtering reinforces inequality by favoring the structurally advantaged influencer and, in most cases, preventing the “competing” influencer from remaining visible to the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Moreover, even in structurally balanced conditions, personalization can lead to the emergence of echo-chambers, in which users’ opinion also radicalizes along non-reference directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The proposed model is a preliminary attempt to describe the complexity of online interactions and comes with some limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In our model, users are passive entities, and influencers are stubborn agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Moreover, homophily is the only driver of individuals’ interaction, as no other relationship structure was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Nonetheless, despite the simplifying assumptions, the emergent behavior of the model proved 29 rich enough to reveal the effects of content personalization and shed light on influencer popularity dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Our work points to several research directions, such as viewing users as active agents capable of publishing their own posts and forwarding (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', sharing) posts from influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This can pose significant challenges in terms of analytic tractability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Another promising direction could be to look at influencers as “strategic” players aiming at maximizing their popularity on the platform by exploiting the internal mechanisms of the platform itself (such as algorithmic content filtering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Description of the dataset We collected data from real online social networks to support the hypotheses of our model and compare emergent behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We focus on two popular social networks: Facebook (FB) and Instagram (IG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Facebook has long been the most popular social media application, while Instagram has undergone a surge in popularity in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In Facebook and Instagram, a profile, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', a social network user, can be followed by other profiles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', its followers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' A profile with a large number of followers is also called an influencer - we consider profiles with more than ten thousand followers as influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Influencers post content (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', posts) consisting of a photo, a video, plain text, or a combination of these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The profile’s followers and anyone registered on the platform can see, like and comment on the influencer’s posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Note that when we use the term influencer, we do not only mean individuals but also groups, soccer teams, newspapers, or companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In this work, we are interested in the plain-text messages of influencers, their temporal sequence, and metadata describing the features of the influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' To get the list of such popular profiles, we exploited the online analytics platform hypeauditor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='com for IG, and www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='socialbakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='com and www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' pubblicodelirio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='it for FB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We restricted the analysis to influencers with at least 10, 000 followers on June 1, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The lists of 649 influencers we used are publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5 For each monitored profile, we downloaded the corresponding metadata, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', the profile information and all generated posts, using the CrowdTangle tool and its API6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' CrowdTangle is a content discovery and social analytics tool owned by Meta and available to researchers and analysts worldwide to support research, subject to a partnership agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' For each influencer, we downloaded all the data related to the posts published between January 1, 2016, and June 1, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Finally, we stored the data, which takes around 110 GB of disk space, on a Hadoop-based cluster, and we used PySpark for scalable processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Details on post classification One of the novelties introduced in this work is the concept of reference direction, which states that influencers have a preferred topic of discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' To confirm this hypothesis, we developed a classifier that can categorize posts according to a particular set of subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' First, we arbitrarily identified a subset of topics that sufficiently characterize the discussions on the monitored profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Specifically, these topics are sports, politics, food and cooking, music, and pandemics, which are intentionally loose and relatively uncorrelated to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We developed a keyword classifier to classify the posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' For each topic, we manually defined a list of representative keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' For example, if we consider pandemic, we search for words like COVID, pandemic, and coronavirus in Italian (and commonly used terms in other languages).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We search for the topic-specific terms in the text corpus of the post, and if we find a match, we mark the post as belonging to the topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Notice that since keywords of various topics may be present in the same corpus, we can flag a 5https://mplanestore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='polito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='it:5001/sharing/P4WnRClQn 6https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='com/CrowdTangle/API 30 message as discussing multiple topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In this work, we discard posts marked as multiple and only consider posts associated with a single topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We are not interested in classifying all posts by an influencer, first because our list of topics does not cover all possible ones, and second because we only need a large enough subsample of posts to make some statistical considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Conversely, it is of utmost importance that the accuracy of the classifier is high since misclassified posts could lead to wrong conclusions about the distribution among the available topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Therefore, we manually validate the accuracy of our methodology for topic detection, as described in the following paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Classifier Precision Evaluation We empirically evaluated the accuracy of the classifier by taking a random subsample of the labeled posts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=', 100 posts for each topic for a total of 500 messages, and manually classifying them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' To this end, we defined a lower and upper bound for accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Indeed, even for a human being, it is challenging to univocally classify posts based on their content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Therefore, we defined three possible states for each classification decision: “t” correct classification, “f” incorrect classification, and “ncc” standing for not completely correct (indicating that the assigned topic is related to the post but may not be the main topic of the post or the classification of the post is difficult).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Given this states subdivision, the precision bounds are as follows: PL = Nt Nt + Nf + Nncc (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1) PU = Nt + Nncc Nt + Nf + Nncc (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2) We refined our term selection for each topic to improve precision based on this analysis .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='7 The classifier’s precision is subject-dependent but was consistently above 80% considering the upper bound defined in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The classification is particularly effective in the case of politics and pandemic, where the precision goes above 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1 summarises the bounds on precision achieved by the procedure described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' These results are sufficient to use the classification to support our modelling assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1: Per-topic Precision Topic Precision l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Precision u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Sports 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='9 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2 Politics 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='0 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='4 Music 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='4 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5 Food 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='5 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='4 Pandemic 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='6 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1 The average percentage of messages classified is 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='8% for all influencers in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Considering the final classifier and the analysed dataset, we automatically flagged about one million posts8 with at least one topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Of these, only 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='7% of the posts were flagged with multiple labels, indicating the message dealt with more than one topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We decided to consider in the rest of the work only influencers for whom it was possible to classify more than a thousand posts in the observed period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' At the end of this filtering process, we could keep 237 influencers for whom the average posts’ classification percentage is 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The dataset used contains a subset of Italian politicians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' To check the correctness of the labelling procedure, we checked whether the derived reference topic for all politicians was politics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' It turned out that 7We make the final list of terms available at https://mplanestore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='polito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='it:5001/sharing/0wD5oU6xr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 81167963 posts were tagged with at least one label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' 31 two politicians did not have politics as reference: Vincenzo De Luca had pandemic, and Renata Briano had food.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' However, this is entirely understandable as the latter runs a food blog and the former was known for his firm and frequent statements on the pandemic situation during the COVID -19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Proofs of Theorems (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2) Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Proof of Theorem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1) Let us start assuming ki(0) > 0 ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In such a case we denote with i0 = arg maxi ki(1) and with K := ki0(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' First we show that the problem: ki(yi) − cyi = 0, with yi ∈ [0, 1] ∀i (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1) admits a solution for any c ≥ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Indeed by choosing c ≥ K we have that necessarily ki(1) ≤ ki0(1) ≤ c·1 ∀i while ki(0) > c · 0 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' therefore a zero zi(c) must exist for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' This zero is unique as a consequence of the concavity of ki(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' The set of zeros zi(c)i provides a solution of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Now to get a solution of the original problem (12) we need to show that there exist a c such that {zi(c)}i are normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Observe that for c = K by construction zi0(K) = 1 while 0 < zi(K) ≤ 1 for i ̸= i0, therefore � i zi(K) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Now, due to the monotonicity and concavity of ki(·), zi(c) is by construction decreasing with respect to c, moreover zi(c) → 0 as c → ∞ ∀i, therefore since � i zi(·) is a continuous function of its argument, there will necessarily be a c0 in correspondence of which � i zi(c0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In the case in which ki(0) = 0, observe that 0 is a solution of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1) for any c, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' zi(c) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Moreover for any c ≥ K a second zero may exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' For example, by construction, zi0(K) = {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Therefore for c = K, as before, we can always choose as set of zeros {zi(K)}i, such that zi(K) = 0 if ki(0) = 0, and i ̸= i0, zi0(K) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' By construction � i zi(K) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In particular � i zi(K) > 1 is there exists a i such that ki(0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' In this latter case, by increasing c all the non null zeros decrease, therefore, as before, there will necessarily be a c0 in correspondence of which � i zi(c0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' □ Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Proof of Theorem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='2) We first show that ||¯π(1) − ¯π(2)||L∞ = maxi |¯π(1) i − ¯π(2) i | = ||G(F1(x, z)) − G(F2(x, z))||L∞ ≤ M||F1(x) − F2(x)||L∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' then we show that we can always enforce: ||F1(x, z) − F2(x, z)||L∞ = ||H(¯π(1)) − H(¯π(2))|| ≤ 1/(2M)||¯π(1) − ¯π(2)||L∞ by properly choosing ω(·, ·) and θ(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Therefore, we can conclude that ||H ◦ G(F1(x, z)) − H ◦ G(F2(x, z))|| ≤ 1/(2M)||(G(F1(x)) − (G(F2(x))|| ≤ M/(2M)||F1(x, z) − F2(x, z)|| = 1/2||F1(x, z) − F2(x, z)||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' First note that ||F1(x, z) − F2(x, z)||L∞ = supx |F1(x, z) − F2(x, z)| coincides with the Kolmogorov distance between the two distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Let us denote with ki(y, F1(x, z)) = λf (i) � � θ(|x − xi|)ρ(¯πi, |x − xi|)dF1(x, z), and similarly for ki(y, F2(x, z)) we assume that: sup y∈[0,1],i |ki(y, F1(x, z)) − ki(y, F2(x, z))| := ∆K(F1, F2) ≤ a||F1(x, z) − F2(x, z)||L∞ a ∈ R+ and dki(y, F1(x, z)) dy |y=0< max i ki(1, F1(x, z)) dki(y, F2(x, z)) dy |y=0< max i ki(1, F2(x, z)) ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Without lack of generality we assume maxi ki(1, F1(x)) ≥ maxi ki(1, F2(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Let the pair (¯π(1) = {¯π(1) i }i, c1) be the solution of ki(yi, F1(x, z)) − cyi = 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='t � i yi = 1, yi ≥ 0, ∀i 32 now let ({�p(2) i }i) the non necessarily normalized solution of ki(yi, F2(x, z)) − c1yi = 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='t yi ≥ 0, ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' by means of elementary geometric considerations we can bound: |¯π(1) i − �p(2) i | ≤ ∆K(F1, F2) c1 − h1 where h1 = dki(y,F1(x)) dy |y=min(¯π(2) i ,�p(2) i )≤ dki(y,F1(x)) dy |y=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' We recall that by construction (see proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='1) we have c1 > maxi ki(1, F1(x))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Denoting with |�p(2)| = � i �p(2) i , we have 1 − � i |�p(2) i − ¯π(1) i | ≤ |�p(2)| ≤ 1 + � i |�p(2) i − ¯π(1) i | Now denoted with ({¯π(2) i }i, c2) the solution of ki(yi, F2(x, z)) − cyi = 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='t � i yi = 1, yi ≥ 0, ∀i we have, by construction, that: 1 max(1, |�p(2)|) < c1 c2 < 1 min(1, |�p(2)|) and therefore, exploiting again elementary geometrical arguments, we can bound: |�p(2) i − ¯π(2) i | ≤ ���� �c1 − h2 c2 − h2 − 1 � �p(2) i ���� where h2 = dki(y,F2(x,z)) dy |y=min(�p(2) i ,¯π(2) i )= dki(y,F2(x,z)) dy |y=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Putting everything together, we have proved that: max i |||¯π(1) i − ¯π(2) i || = ||G(F1(x, z)) − G(F2(x, z))||L∞ ≤ M||F1(x, z) − F2(x, z)||L∞ To conclude the proof, first note that by properly choosing ρ(·, ·) and θ(·) we can assume vx(x, z) and σ2 x(x, z) to depend sufficiently smoothly on ¯π, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' ∀ε > 0 we can assume: sup x ��� ���v(1) x (x, z) − v(2) x (x, z) ��� ��� L∞ ≤ ε||¯π(1) − ¯π(2)||L∞ ∀z, sup x ��� ���σ2,(1) x (x, z) − σ2,(2) x (x, z) ��� ��� L∞ ≤ ε||¯π(1) − ¯π(2)||L∞ ∀z, and sup x ����� ����� ∂σ2,(1) x (x, z) ∂x − ∂σ2,(2) x (x, z) ∂x ����� ����� L∞ ≤ ε||¯π(1) − ¯π(2)||L∞ ∀z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' Then observe that the solution of the Fokker-Planck equation given in (9) on a compact interval (and so also its primitive) depends smoothly on function vx(x, z), function σ2 x(x, z) and its first derivative, as long as infx,z σ2 x(x, z) is bounded away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' As final remark note that the set of weakly-increasing functions F(x), such that F(a) = 0 and F(b) = 1 equipped with the L∞-norm forms a closed set in a complete metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfg_yi/content/2301.01478v1.pdf'} +page_content=' □ 33 References [1] M.' 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