diff --git "a/8tAyT4oBgHgl3EQfQ_YL/content/tmp_files/load_file.txt" "b/8tAyT4oBgHgl3EQfQ_YL/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/8tAyT4oBgHgl3EQfQ_YL/content/tmp_files/load_file.txt" @@ -0,0 +1,2653 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf,len=2652 +page_content='A Bayesian latent position approach for community detection in single- and multi-layer networks with continuous attributes Zhumengmeng Jina,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Juan Sosab and Brenda Betancourtc a University of Florida b Universidad Nacional de Colombia c NORC at the University of Chicago December 2022 Abstract The increasing prevalence of multiplex networks has spurred a critical need to take into account po- tential dependencies across different layers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' especially when the goal is community detection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' which is a fundamental learning task in network analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' We propose a full Bayesian mixture model for community detection in both single-layer and multi-layer networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' A key feature of our model is the joint modeling of the nodal attributes that often come with the network data as a spatial process over the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' In addition, our model for multi-layer networks allows layers to have different strengths of dependency in the unique latent position structure and assumes that the probability of a relation between two actors (in a layer) depends on the distances between their latent positions (multiplied by a layer-specific factor) and the difference between their nodal attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Under our prior specifications, the actors’ positions in the latent space arise from a finite mixture of Gaussian distributions, each corresponding to a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Simulated examples show that our model performs favorably compared to the existing ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' The model is also applied to a real three-layer network of employees in a law firm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 1 Introduction Network data conveniently describes the relationships between actors in complex systems and is ubiquitous in many statistical applications, including finance, social science, criminology, biology, epidemiology, and computer science, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Understanding the relationships between actors can aid domain experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' multiplex network, community detection, latent position model, mixture model, spatial process, visu- alization 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='00055v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='AP] 30 Dec 2022 For instance, in epidemiology, people in a certain area can be portrayed in a contact network that can be studied to detect infectious disease outbreaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' In criminology, communications between terrorists form a terrorist network, helping intelligence agencies to better counter terrorism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Many models have been developed for the inference of networks over the past decades (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=', Erdös and Rényi, 1959, Frank and Strauss, 1986), among which the broad class of latent space models is one of the most widely used (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=', Sosa, 2021 for an exhaustive review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Suppose the network under study has N actors, then under latent space models, there are N independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=') latent variables z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' , zN, one for each actor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Under a mild exchangeability assumption in Hoff [2007], results in Aldous [1985] and Hoover [1982] show that edge variables yi,j depend on latent variables through a symmetric function γ(zi, zj) that is meant to capture any pattern in the network beyond any known covariate information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Many well-known models fall into the category of latent space models, which can be distinguished between two cases depending on whether latent variables are discrete or continuous [Matias and Robin, 2014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' For in- stance, stochastic block models [Nowicki and Snijders, 2001, Wang and Wong, 1987] – hereafter SBM – are special cases of latent space models with discrete latent variables zi ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' , K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' When latent variables are assumed to be continuous, another approach using latent variables is the class of latent position models (LPM) proposed by Hoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' [2002] which our model in the paper is built upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' In its basic formulation, LPMs model the edge variables yi,j as conditionally independent given the distance between latent variables γ(zi, zj) = −∥zi − zj∥, which naturally accounts for transitivity effects through the latent space (typically a Euclidean K-dimensional space for a predetermined K) where zi lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Later on, Handcock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' [2007] proposed an extension on Hoff et al.’s LPM, namely the latent position cluster model (LPCM), by imposing a Gaussian mixture prior on the latent positions to perform clustering tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Krivitsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' [2009] further extended Handcock et al.’s model by adding the random sender and receiver effects proposed by Hoff [2005].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Other formulations of γ(·, ·) can be found in Schweinberger and Snijders [2003], Hoff [2005, 2009], Athreya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' [2017], Minhas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' [2019], among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Besides edge information of a network, extra information like node and edge attributes and different types of edges are often available, and should ideally be leveraged for inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Typical ways to incorporate attributes in a network model include: (1) modeling the network as a function of the attributes (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=', Hoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=', 2002, Hoff, 2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' (2) modeling the attributes as a function of the network [Guha and Rodriguez, 2021];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' (3) jointly modeling the network and attributes (Linkletter, 2007, Kim and Leskovec, 2012, Fosdick and Hoff, 2015, Ciminelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' The first approach is arguably the most common approach to incor- porate covariates into the model, but we consider an approach of joint modeling proposed by Ciminelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 2 [2019], namely the social network spatial model (SNSM), where the authors modeled edges yi,j as condi- tionally independent given ∥zi − zj∥ and the distance of the continuous node attributes ∥xi − xj∥, and node attributes are further modeled as a spatial process over the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Note that joint modeling does not require the network or the attributes to be fully observed as the first two approaches, hence one could predict missing network and attribute data (if there is any).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' In addition, it improves model fitting by capturing the dependence structure between latent variables and the attributes (when such dependency exists), as we will see in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' We propose a full hierarchical Bayesian model that builds on Ciminelli et al.’s SNSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Instead of using a Gaussian distribution as the prior for latent positions as in Ciminelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' [2019], we impose a Gaussian mixture prior as in Handcock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' [2007], so that our model could also capture the group structure in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Detecting communities or clusters among actors in the network is an important task in network analysis and has spurred the development of many models and algorithms, among which the SBM has motivated an active line of research that deals with community detection (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=', Lee and Wilkinson [2019] for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' However, SBM may not fit well when many actors fall between clusters [Hoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=', 2002].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' We will compare our model with an SBM that incorporates covariates as fixed effects (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=', model the edge variables as a function of latent classes and covariates [Leger, 2016]), and we call this model a covariate-assisted stochastic block model (CSBM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' We will show that our model presents improved model fitting while producing similar clustering results as CSBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' We also propose an extension of our model to multi-layer network settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Multi-layer networks can gen- erally be categorized into two cases: cross-sectional networks that have different types of connections (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=', social networks of friendship, coworker-ship, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=') and time-varying networks where the same type of con- nections are measured over time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=', a trade network that changes over time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' We consider a type of cross-sectional multi-layer network where each layer has a common set of actors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Substantial work has been done on latent space models for cross-sectional multi-layer networks that take a Bayesian approach (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=', Gollini and Murphy, 2016, Salter-Townshend and McCormick, 2017, D’Angelo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=', 2019, Sosa and Betan- court, 2022, Durante and Dunson, 2018, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=', 2019, MacDonald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' In extending our model to the multiple networks setting, we adopt the approach in Sosa and Betancourt [2022] in a parsimonious way, where latent positions are assumed to be the same for all layers, but the strength of borrowing such latent structure information is allowed to be different across different layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Note that, the original model in Sosa and Betancourt [2022] assumed different latent positions for different layers and had an additional hierarchy on the hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' The specification of our model is given in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Section 2 contains general background on the spatial 3 process and introduces the proposed model (for single- and multi-layer network settings) which we call the latent position joint mixture model (LPJMM) in the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' In addition, prior specification, identifiable problem, and inference will also be discussed in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Several simulation studies are conducted in section 3, where LPJMM is compared with Handcock et al.’s LPCM, Ciminelli et al.’s SNSM and CSBM in single-layer settings and the model is also evaluated in multi-layer settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' In section 4, we apply LPJMM to a real-world multi-layer network data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Finally, we conclude with some discussion in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 2 Models We first review the LPM introduced in Hoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' [2002], and then build upon it with a spatial process to allow for joint modeling of the network and the nodal attributes, and with a finite Gaussian mixture distribution for latent positions to allow for clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Consider a binary single-layer network with N actors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Denote its adjacency matrix as Y = (yi,j) ∈ {0, 1}N×N, where yi,j = 1 if actors i and j are connected, and yi,j = 0 if they are not connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Suppose the network data comes with a one-dimensional nodal attribute xi for each actor, and denote the covariate as x = (xi) ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' The LPM assumes that each actor i has an observed latent position zi in a K-dimensional Euclidean latent space, the so-called latent space, for some K ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Let z = (zi) ∈ RN×K, then LPM models edge yi,j as conditionally independent given distances between nodal attributes as well as distances between latent positions via logistic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' But instead of the logistic link, we use the probit link in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' The analysis of probit regression models can often be facilitated by a Gibbs sampler constructed using the data augmentation approach that introduces latent variables with truncated normal distributions [Albert and Chib, 1993].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' (See also Sosa and Betancourt (2022) for a discussion on the choice of link functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=') Specifically, for i, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' , N} and i ̸= j, yi,j | z, x, a, b, θ ind ∼ Ber � Φ(a + b|xi − xj| − θ∥zi − zj∥) � , (1) where a, b ∈ R and θ ∈ R+, Ber(p) is a Bernoulli distribution that takes value 1 with some probability p, ∥ · ∥ is the Euclidean norm on RK and Φ(·) is the cumulative distribution function of the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Note that we impose a factor θ for the distance between latent positions, which is different from Hoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' [2002] and Krivitsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' [2009].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Although θ is unidentifiable in single-layer networks, it plays a non-trivial role in multi-layer network settings (introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' We defer a detailed discussion of θ to Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 4 To allow for joint modeling of the network and nodal attributes, we model the nodal attributes as a spatial process over the latent space RK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Hence, nodal attributes are treated as random variables indexed by their latent positions, and the distance between these random variables is found by the distance between their corresponding positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' As in Ciminelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' [2019], we specify the spatial process as a Gaussian process that is stationary with mean β and isotropic (see Banerjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=', 2015 for definitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' In this case, the process is completely defined by its covariance function Cov(d), where d is the distance between two random variables in the Gaussian process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' In particular, we specify Cov(d) with an exponential kernel, that is, Cov(d) = � � � � � τ 2 + σ2, if d = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' σ2 exp(−φd), if d > 0, where τ ≥ 0, σ > 0 and φ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' It is well-known that such a covariance structure is valid, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=', the covariance matrix for any finite collection of random variables in the process is positive definite [Banerjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=', 2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Let Mz = (mij) ∈ RN×N where mij = exp(−φ∥zi − zj∥) and denote IN as the N-dimensional identity matrix, then the Gaussian process of the nodal attributes is constructed as follows, x | z, β, σ, τ, φ ∼ NN(β111N, σ2M(z, φ) + τ 2IN), (2) where Nd is a d-dimensional multivariate normal distribution for some dimension d ∈ {2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' }, and 111N is an N-dimensional vector with all 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' As in Krivitsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' [2009], we impose a Gaussian mixture distribution on latent positions, which allows us to cluster actors into different groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Suppose there are H < ∞ predetermined number of components in the Gaussian mixture distribution, then zi | ωωω,µµµ,κκκ ind ∼ H � h=1 ωhNK(µh, κ2 hIK) , (3) where ωωω = {ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' , ωH}, µµµ = {µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' , µH}, κκκ = {κ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' , κH}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Note that µh is a K-dimensional mean vector where h ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' , H}, and ωh is the probability that an actor belongs to the h-th group such that ωh ∈ (0, 1) and �H h=1 ωh = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' In single-layer network settings, the model is given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' (1) to (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Under our model, nodal attributes of two actors whose latent positions are close are more likely to be similar according to the exponential covariance structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' If b < 0 (b > 0), actors with similar attributes are more (less) likely to be connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' When b = 0, nodal attributes do not affect the distribution of the network directly (but it still has an indirect 5 Figure 1: DAG representation of the LPJMM in multi-layer settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' impact on the network through latent positions by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='1 An extension to multi-layer networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Our model can also be extended to multi-layer network settings in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Suppose we have L layers Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' , YL in the network, where all layers are defined over the same set of actors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' We assume the same latent positions z for all layers but allow the strength of borrowing such latent structure information to be different by imposing layer-specific factors θℓ for ℓ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' , L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Our model in multi-layer settings is then presented as follows yi,j,ℓ | z, x, aℓ, bℓ, θℓ ind ∼ Ber � Φ(aℓ + bℓ|xi − xj| − θℓ∥zi − zj∥) � , (4) x | z, β, σ, τ, φ ∼ NN(β111N, σ2M(z, φ) + τ 2IN) , (5) zi | ωωω,µµµ,κκκ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' ∼ H � h=1 ωhNK(µh, κ2 hIK) , (6) where yi,j,ℓ is the edge variable between actors i and j in layer ℓ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' , L}, aℓ, bℓ and θℓ are layer-specific parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Note that Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' (5) and (6) are the same as Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 1 shows a directed acyclic graph (DAG) representation of the model given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' (4) to (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='2 Prior specification We take a Bayesian approach to estimate the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Without loss of generality, a Bayesian ver- sion of the model given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' (4) to (6) is formed by placing prior distributions on the unknown parameters aℓ, bℓ, θℓ, β, σ, τ, φ, ωωω, µµµh, κh, for ℓ = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' , L} and h = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' , H}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' In the model we consider, these parameters are assumed a priori independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' For parameters in the probit regression tier as specified by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' (4), their priors are specified as follows: aℓ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' ∼ N(ma, ν2 a) , bℓ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' ∼ N(mb, ν2 b ) , θℓ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' ∼ Gamma(λ1, λ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' The priors for the parameters in the spatial process tier as given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' (5) are given as follows: β ∼ N(0, ν2 β) , σ2 ∼ InvG(η1, η2) , τ 2 ∼ InvG(ξ1, ξ2) , φ ∼ U(u1, u2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Finally, we put the following priors on the rest of the parameters: ωωω ∼ Dir(α) , µh i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' ∼ NK(mµ, ν2 µIK) , κ2 h i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' ∼ InvG(γ1, γ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Note that, ma, νa, mb, νb, λ1, λ2, νβ, η1, η2, ξ1, ξ2, u1, u2, α, mµ, νµ, γ1 and γ2 are user-specified hyperparameters, and Gamma(·, ·), InvG(·, ·), U(·, ·), Dir(·) represents Gamma, Inverse-Gamma, uniform, and Dirichlet distributions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='3 Posterior distribution and model estimation As is standard in Bayesian estimation of mixture models (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=', Diebolt and Robert [1994]), we define a new variable gi that serves as the missing data of group membership of actor i whose distribution depends on ωωω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' In particular, gi = h if actor i belongs to the h-th group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' The joint density of (zi, gi) given ωωω, µµµ and κκκ is then given by H � h=1 � ωh 1 � 2πκ2 h exp � − 1 2κ2 h ∥zi − µh∥2��I{gi=h} , where the indicator function I{gi=h} = 1 if gi = h, and I{gi=h} = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Let g = (gi)N i=1 be the group membership for all actors and L(·) be the law of a random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Then the posterior distribution of z, g 7 and the parameters upon which priors are specified in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='2 is given by Π(z, g, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' , aL, b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' , bL, θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' , θL, β, τ 2, σ2, φ,ωωω,µµµ,κκκ | Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' , YL, x) ∝ � L � ℓ=1 L(Yℓ | z, x, aℓ, bℓ, θℓ) � L(x | z, σ, τ, φ)L(z, g | ωωω,µµµ,κκκ) � L � ℓ=1 L(aℓ)L(bℓ)L(θℓ) � × L(β)L(σ2)L(τ 2)L(φ)L(ωωω)L(µµµ)L(κκκ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Note that the dimension of the posterior distribution has dimension NK + N + 3L + 3H + 4 and the corresponding posterior density is presented as follows, π(z, g, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' , aL, b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' , bL, θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' , θL, β, τ 2, σ2, φ,ωωω,µµµ,κκκ | Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' YL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' x) ∝ N � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='j=1 i̸=j L � ℓ=1 � Φ(aℓ + bℓ|xi − xj| − θℓ∥zi − zj∥) �yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='ℓ� 1 − Φ(aℓ + bℓ|xi − xj| − θℓ∥zi − zj∥) �1−yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='ℓ × |σ2M(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' φ) + τ 2IN|− 1 2 exp � − 1 2(x − β1)⊺� σ2M(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' φ) + τ 2IN �−1(x − β1) � × N � i=1 H � h=1 � ωh � κ2 h exp � − 1 2κ2 h ∥zi − µh∥2��I{gi=h} × exp � 1 2ν2a L � ℓ=1 (aℓ − ma)2 + 1 2ν2 b L � ℓ=1 (bℓ − mb)2� L � ℓ=1 θλ1−1 ℓ exp(−λ2θℓ) × exp � β2 2ν2 β � (σ2)−η1−1(τ 2)−ξ1−1 exp � − η2 σ2 − ξ2 τ 2 � I{φ∈[u1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='u2]} × H � h=1 � ωαh−1 h I{�H h=1 ωh=1} exp � − 1 2ν2µ ∥µh − mµ∥2� (κ2 h)−γ1−1 exp � − γ2 κ2 h �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='4 Inference and identifiability of parameters Note that the posterior distribution is highly intractable, hence we must resort to Markov chain Monte Carlo (MCMC) methods for inferences on model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' A Markov chain of the parameters is generated via the program “Just Another Gibbs Sampler” (JAGS) which is implemented in R [R Core Team, 2021] using the rjags package [Plummer, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Several parameters are not identifiable in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Firstly, due to factors θℓ and φ, and the fact that latent positions are incorporated in the posterior only through their distances, the posterior is, therefore, invariant to θℓs and φ, and is invariant to scaling, reflection, rotation, and translation of the latent positions z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' (Note that, Hoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=', 2002 and Krivitsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=', 2009 did not have θℓs, hence their posterior is not invariant to the 8 scaling of latent positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=') Although θℓs are not identifiable and do not affect the model fitting, in multi- layer settings, their ratios θℓ1/θℓ2 still provide valid information on layer’s relative strength of borrowing information from the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Despite being unidentifiable, one can still make inferences on the latent positions and find a reasonable estimate for z through a post-process which we now describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Similar to the definition in [Hoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=', 2002], we define the equivalence class of z ∈ RN×K, denoted as [z], to be the set of positions that are equivalent to z under scaling, reflection, rotation, and translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Given a fixed reference position zref, a position z∗ is found in [z] such that z∗ = arg minz′∈[z] tr(zref − z′)⊺(zref − z′), which is the so-called Procrustes transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' In simulation studies, zref is naturally chosen to be the true latent position, while in practical applications, we could use the last iteration of the Markov chain of latent positions as the reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' The Procrustes transformation is performed for each iteration of the Markov chain of the latent positions {zn}, and an estimate for z is taken as the mean of the Procrustes transformations of {zn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' As occurs in Bayesian mixture models, the label-switching problem for the group membership g is an- other source of non-identifiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' That is, the posterior is invariant under permutations of clustering labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Many algorithms have been proposed to obtain a single clustering estimate based on the MCMC sample of the group membership {gn},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' including an optimization method (which we call “MaxPEAR” hereafter) in Fritsch and Ickstadt [2009] that finds a clustering that maximizes posterior expected adjusted rand index,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' an optimization method (“MinBinder”) in Lau and Green [2007] that minimizes Binder’s loss function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' and a greedy algorithm (“GreedyEPL”) in Rastelli and Friel [2018] that aims to minimize the variation of informa- tion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' These approaches may generate different clustering estimates, and to get a better under- standing of the model performance, all aforementioned algorithms (MaxPEAR, MinBinder and GreedyEPL) are used to assess the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Estimates based on these approaches are found using the packages GreedyEPL [Rastelli, 2021] and mcclust [Fritsch, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 3 Simulation Two simulation studies are carried out in this section to evaluate our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' A single-layer network is considered in the first simulation where we compare LPJMM with three other models designed only for single-layer networks, namely LPCM in Handcock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' [2007], SNSM in Ciminelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' [2019], and CSBM in Leger [2016], where SNSM is also implemented using the rjags package, and LPCM and CSBM are implemented using the latentnet [Krivitsky and Handcock, 2022] and sbm [Chiquet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=', 2022] packages respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' The model specifications for these models can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Models assessments include how well a model could recover the group membership and the latent position configuration, and a 9 1 Figure 2: Left: A visualization of the network based on the true latent position and color indicates group membership g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Right: Heatmap of the adjacency matrix (where actors are reordered according to g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' goodness-of-fit test using summaries of networks including density, transitivity, and assortative coefficient with respect to the group membership g (see Kolaczyk and Csárdi, 2020 for definitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' We also evaluate our model by how accurately it could estimate certain parameters in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' The second simulation is conducted in two-layer network settings, where the performance of our model could be further evaluated by how well the ratio θ1/θ2 can be recovered that reflects differences in each layer’s dependency on the latent position structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='1 Simulation 1: a single-layer network Consider a single-layer network (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=', L = 1) with N = 100 actors generated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Firstly, generate latent positions z from a mixture of H = 5 multivariate normal distributions, and then generate attributes x jointly from a multivariate normal distribution with mean β1N = 0 and covariance matrix given by Cov(d) in Section 2 where φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='5, τ 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='3, σ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Finally, the network data is generated according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' (1) with a = 5, b = −2, and θ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 2 for a visualization of the simulated network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' The network is fairly sparse with a density equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='1531, and shows fairly strong transitivity and assortative mixing with coefficients 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='5049 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='5512 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' As for the prior specifications, we set ma = mb = 0, and ν2 a = ν2 b = 9 to allow a wide range of values for a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Let θ ∼ Gamma(1, 1) so that θ has mean 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' An almost flat prior is imposed on β by setting νβ = 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' The same uniform prior U(0, 1) as in Ciminelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' [2019] is specified for φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' We suggest the sum of the prior means of τ 2 and σ2 to be on the same scale as the sample variance of x, and here we use σ2 ∼ InvG(2, 1) and τ 2 ∼ InvG(2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Let α = 1 so that the prior on ωωω is a flat Dirichlet distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Following the heuristics in Sosa and Betancourt [2022], we specify µh i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' ∼ NK(0, 2/3IK) and κ2 h i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' ∼ InvG(3, 2/3) so that var(zij|gi) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 10 our model name LPCM CSBM MaxPEAR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='737 (5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='707 (4) – MinBinder 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='712 (11) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='748 (10) – GreedyEPL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='664 (4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='688 (4) – Variational-EM – – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='707 (6) Table 1: Adjusted Rand indices corresponding to different estimation methods for group membership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Num- bers in the parentheses represent numbers of estimated groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='1 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='98 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='99 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='(d) CSBM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='Figure 3: (A): Color indicates the true group membership g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' (B)-(D): Color indicates the estimated group memberships ˆg of LPJMM, LPCM and CSBM respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Positions of the points in all plots are true latent positions z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Note that the latent space dimension K and the number of clusters H in the model need to be prespecified along with the priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' We take K to be the true dimensions of the latent space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=', K = 2) since this facilitates model assessment by allowing visualizations of the estimated latent positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' One could also use the Watanabe-Akaike Information Criterion (WAIC) to select a K with the smallest WAIC as in Sosa and Betancourt [2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' However, WAIC and other information criteria like Deviance Information Criterion (DIC) are not helpful in choosing the number of clusters H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' We noticed that the model assessment is significantly worse when H is chosen to be smaller than the truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' However, model assessments are similar among models whose H is at least as large as the truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' A comparison of the model assessment for different specified H is given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' From the comparison, we could also see that when H is specified to be larger, the number of clusters in the estimated group membership ˆg also tends to be larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Therefore, we suggest choosing H to be the largest number of groups that one is willing to accept, and in this example, we choose H to be 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' We then fit LPJMM using MCMC sampling with 20 000 burn-in iterations and a further 10 000 iterations which are kept for posterior analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' The Markov chain mixes reasonably well and shows no signs of lack of convergence (see Appendix C for the traceplot of the log-likelihood chain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 11 LPCM LPJMM (simulation 1) LPJMM (simulation 2) Sum of Euclidean distances 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='08 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='06 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='20 Table 2: Sum of distances between the estimated and true latent positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' To evaluate a model’s ability to recover the group membership, we first find estimates of clustering using the optimization algorithms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=', MaxPEAR, MinBinder and GreedyEPL) mentioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' The adjusted Rand index is then calculated for each clustering estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Note that SNSM does not define clusters, therefore we only compare the adjusted Rand index between LPJMM, LPCM, and CSBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Since the sbm package takes a non-Bayesian approach that uses a Variational-EM algorithm to find a point estimator for the group membership g, optimization methods like MaxPEAR are not necessary to analyze results from CSBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' The results shown in Table 1 suggest that these three models have a similar ability in recovering group membership, with rand indices of LPJMM using the MaxPEAR and MinBinder algorithms being higher than the rand index (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='707) under the CSBM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' A visualization of the estimated clusters based on the true latent positions is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Also, notice that the MinBinder algorithm tends to overestimate the number of clusters in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' To further compare the ability to recover latent position configuration between LPJMM and LPCM, we find an estimate of the latent positions as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Firstly, we perform the Procrustes transformation on zn for each iteration n, and then take the estimate ˆz of z to be the average of zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' We then calculate the Euclidean distance between the estimated latent position ˆzi (which is the i-th row in ˆz) and the true latent position zi (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=', the i-th row in z) for each actor i and use the sum of distances of all actors to quantify the similarity between the estimated and the true latent position configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' The results are shown in Table 2 which suggests that these two models have similar recovery of the latent positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Plots of the estimated latent positions of LPJMM and LPCM can be found in Appendix D, which also suggest similar estimated configurations of z as Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Following Sosa and Betancourt [2022], we assess if models have a good fit in the sense of good reproduction of a variety of summary statistics, which are calculated based on a collection of simulated networks generated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' For LPJMM and SNSM, a network is simulated for every 10-th iteration using the parameters in that iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' For LPCM and CSBM, 1000 networks are simulated using their respective packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Then for each model, we calculate the density, transitivity, and assortative coefficient (if applicable) with respect to the true group membership for each simulated network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Boxplots of these summary statistics are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 4 and the averages of these summary statistics for each model are given in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Note that our model 12 density transitivity assortativity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='60 LPJMM LPCM SNSM CSBM Figure 4: Boxplots of summary statistics for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Red dotted lines indicate the true values for network characteristics respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' true value LPJMM LPCM SNSM CSBM density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='1531 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='1539 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='1530 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='1504 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='1499 transitivity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='5049 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='5144 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='5467 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='4027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='3776 assortativity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='5512 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='5468 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='5475 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='4811 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='4954 Table 3: Means of the summary statistics of the simulated networks for each model in simulation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' appropriately captures these structural features of the network data, while LPCM tends to overestimate tran- sitivity in the network, and both SNSM and CSBM tend to underestimate both transitivity and assortativity in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='2 Simulation 2: a two-layer network Continue using the parameter setup in simulation 1 and its generated network as the first layer (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=', a1 = 5, b1 = −2, θ1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='72), we generate a second layer of the network with a2 = 3, b2 = 1, θ2 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' As in simulation 1, we fit LPJMM with K = 2 and H = 5 and evaluate the model’s ability to recover the group membership using the adjusted Rand indices based on four clustering summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' The results are given in Table 4, which shows similar clustering estimates as in simulation 1 where only one layer is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' However, the sum of Euclidean distances between the estimated and true latent positions of all actors (see Table 2) in simulation 2 is 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='20, which is a significant improvement compared to 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='06 in simulation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' The plot of the estimated latent position configurations is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 5 (B), which visualizes the model’s recovery of latent positions and group membership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' We also carry out the goodness-of-fit test as in simulation 1 and the result is given in Table 5, which shows that LPJMM captures these structural features accurately, and the result for layer 1 is similar to the result in simulation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 13 MaxPEAR MinBinder GreedyEPL adjusted Rand index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='748 (6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='753 (12) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='662 (4) Table 4: Adjusted Rand indices corresponding to different estimation methods for group membership in simulation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Numbers in the parentheses represent numbers of estimated groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='5 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='(b) Estimated z and g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='Figure 5: (A): Points are plotted based on true latent position z and true group membership g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' (B): Points are plotted using the estimated latent positions in simulation 2, and color represents the estimated group membership using the MaxPEAR method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Recall that θ1 and θ2 are of no direct interest since they are not identifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' However, we are still interested in the ratio θ1/θ2 since it reflects the relative strength of borrowing information from the latent space of each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Although aℓ and bℓ are of no direct interest, we pay attention to their signs, especially that of bℓ because different signs of bℓ have different interpretations of the effect of attributes as discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' We also assess the model’s ability to estimate parameters β, τ 2, and σ2 using posterior means and 95% credible inter- vals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' The results are given in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Overall, the performance of LPJMM in recovering the true values of these model parameters is pretty well, except for τ 2 and σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Both LPJMM and SNSM tend to underestimate σ2 and overestimate τ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' That is, the covariance of the attributes tends to be underestimated, and although τ 2 is slightly overestimated, the variance of the attributes (τ 2 + σ2) still tends to be underestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 4 Real data analysis In this section, we consider a three-layer network data set collected by [Lazega, 2001] from a corporate law firm from 1988-1991 in New England.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' This network describes three types of relationships (namely, networks of advice, friendship, and coworker contacts) between 71 lawyers in the law firm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Several actor attributes are also collected: age, gender, seniority (years with the firm), office (located in Boston, Hartford, or Providence), practice (litigation or corporate law), law school the lawyers attended (Harvard or Yale, University of Connecticut, or other universities) and status (partner or associate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' A principal component 14 true value mean density layer 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='1531 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='1535 layer 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='1024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='1023 transitivity layer 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='5049 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='5088 layer 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='5477 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='5546 assortativity layer 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='5512 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='5466 layer 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='6923 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='6890 Table 5: Means of the summary statistics in simulation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' true value posterior mean 95% credible interval θ1/θ2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='680 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='653 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='579, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='721) a1 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='01 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='719, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='262) a2 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='25 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='976, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='572) b1 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='919 (-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='053, -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='766) b2 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='058 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='901, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='252) β 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='047 (-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='027, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='01 ) τ 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' analysis (PCA) is performed on age and seniority attributes, and the first principal component explains 89% of the variance which is of no surprise since age and seniority are highly correlated with a correlation coefficient being 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' We chose the first principal component to be the attribute x and let H = 8 since it is the largest number of clusters we expect in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Then the model is fitted to the network using the same prior and Markov chain setup as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' The study of the Lazega network in this paper is meant to find out how the three types of relations can be explained by the findings deduced from the model fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' We first visualize the estimated latent positions z colored by different categorical attributes (gender, office, practice, law school, and status) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' As we can see from these plots, the estimated positions z are well separated by the office (especially offices in Boston and Hartford) and practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Compare these plots with z colored by MaxPEAR and GreedyEPL estimated clustering g in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 7, we can see that both estimated g roughly clusters lawyers into three groups: lawyers in Hartford office, litigation lawyers in Boston or Providence offices, and corporate lawyers in Boston or Providence offices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Plots of adjacency matrices of the three layers (where lawyers are grouped by the MaxPEAR estimate of g) and their corresponding networks are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='status ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='Figure 6: Points in all plots are drawn based on the estimated latent positions z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' and are colored based on their categories in gender,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' office,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' practice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' law school,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' and status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' the most estimated clustering pattern, while the advice network presents the least of such pattern, which could also be seen from the relative ratios of θℓs in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' This means that lawyers from the same office and doing the same practice are more likely to become coworkers and friends, but who they seek advice from does not depend much on office and practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Furthermore, we can deduce from the posteriors of bℓ in Table 7 that these lawyers tend to seek advice from people of similar age (or seniority) since the posterior estimate of b1 is negative, while lawyers of different ages (or seniority) are more likely to become friends and coworkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' This conclusion is in line with the assortativity coefficients with respect to the nodal attributes (lawyer’s age) given in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 5 Discussion This paper presents a latent position model that extends LPCM of Handcock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' [2007] and SNSM of Ciminelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' [2019] to jointly model network data and the nodal attributes and perform model-based clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' By jointly modeling the network and the attributes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' we are able to describe how the attributes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='3 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='69 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='71 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='(b) GreedyEPL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='Figure 7: Points are plotted using the estimated latent positions and color indicates the estimated group ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='membership using MaxPEAR and GreedyEPL methods respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' posterior mean 95% credible interval θ1/θ2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='3229 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='2352, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='4152) θ1/θ3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='2035 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='1479, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='2606) θ2/θ3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='6319 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='5536, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='7198) b1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='0986 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='1401, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='0579) b2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='0708 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='0263, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='1137) b3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='133 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='0854, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='186) Table 7: Posterior means and 95% credible intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' change over the network and explain how relations could be influenced by attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' LPJMM also provides an extension to multi-layer network settings on the assumption that all layers share the same latent position structure but with different strengths of borrowing such latent structure information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' We applied our method to two simulated networks, one with a single layer and another with two layers, and found our model to give satisfactory fits to these two data sets and is competitive in terms of goodness-of-fit and group detection compared with SNSM, LPCM, and CSBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' The model is also applied to a three-layer real network data set and we are able to draw reasonable conclusions from the modeling results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' We have suggested choosing the number of groups H to be the largest number of groups that one is willing to accept in the network because we have found that varying the number of groups has almost no impact on the model fit and prediction outcome as long as it is in a reasonable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' One could also fit the CSBM to the network first, and choose H based on its estimated number of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' One problem we have not addressed in the paper is of choosing the dimension of the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' This can be done by using Bayesian model selection like WAIC as in Sosa and Betancourt [2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Our model could be extended in several ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Firstly, other extensions of our model to multi-layer settings could be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' For example, Sosa and Betancourt [2022] assumed conditionally independent layer- 17 advice friendship coworker Figure 8: Upper: Heatmaps of the adjacency matrices (where lawyers are reordered according to the Max- PEAR estimate of g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Lower: A visualization of the three layers based on the estimated z and color indicates the MaxPEAR estimate of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' advice friendship coworker assortativity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='2536 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='1107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='1224 Table 8: Assortativity coefficients with respect to lawyer’s age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' specific latent positions, whereas MacDonald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' [2022] assumed that the latent position of an actor in all layers is (d0 + d1)-dimensional, where the first d0 components of the latent position are the same across all layers, and only the last d1 components are layer-specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Secondly, instead of assigning a user-specified number of groups H to the model, we could learn the number of groups by using a Bayesian nonpara- metric approach with a Dirichlet Process prior to model community memberships (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=', Amini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' LPJMM could also be extended to leverage multivariate covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' So far, we have limited ourselves to mod- eling univariate nodal attributes that are approximately Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' For continuous nodal attributes with more than one dimension, we have used the first principal component from the principal component analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' To take full advantage of high-dimensional nodal attributes, one could use multivariate spatial process modeling to replace Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Other extensions of more sophisticated spatial modeling include spatiotemporal modeling of attributes for time-varying networks, which would help to describe changes in actors over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 18 Appendix A Model Specifications for SNSM, LPCM and CSBM Note that the original SNSM in Ciminelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' [2019] uses the logit link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' In order to make a fair comparison, we also use the probit link in SNSM as in LPJMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' The model specification for SNSM used in this paper is given as follows: yi,j | z, x, aℓ, bℓ, θℓ ind ∼ Ber � Φ(a + b|xi − xj| − ∥zi − zj∥) � , x | z, β, σ, τ, φ ∼ NN(β111N, σ2M(z, φ) + τ 2IN) , and the priors are set to be the same as the priors in LPJMM (if possible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' To be specific, zi i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' ∼ N2(000, I2) , β ∼ N(0, 104) , σ2 ∼ InvG(2, 1) , τ 2 ∼ InvG(2, 1) , φ ∼ U(0, 1) , and the priors on the parameters in the probit regression tier are given by: a i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' ∼ N(0, 9) , b i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' ∼ N(0, 9) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' SNSM in this paper is implemented using JAGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' The model specification for LPCM (see Handcock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=', 2007) is given as the follows, yi,j | z, x, β0, β1 ind ∼ Ber � logit(β⊺ 0xi,j − β1∥zi − zj∥) � , zi | ωωω,µµµ,κκκ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' ∼ 5 � h=1 ωhN5(µh, κ2 hIK) , and we use the default priors given in the latentnet package for prior specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' We first introduce several notations before presenting CSBM in Leger [2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Suppose there are Q groups in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Denote the N × Q group membership matrix as ZZZ = {Ziq}, and Ziq = 1 if actor i belongs to group q, Ziq = 0 if otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' It is assumed that an actor can only belong to one group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' The model specification for CSBM is given as follows, yi,j | Zi, Zj, x, β ind ∼ Ber � logit(mqi,qj + β⊺xi,j) � , where Zi is the i-th row of ZZZ, qi is the group membership for actor i and the group effect mqi,qj ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 19 B Comparing model performances for different number of groups We conduct a comparison of LPJMM with different H ∈ {3, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' , 9} using the data set in simulation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Table 9 presents the adjusted rand indices, and the results are similar for models that assume H to be equal to or larger than the true number of groups (which is 5 in this example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' However, the adjusted rand indices for all three estimates are significantly smaller when the model assumes H to be smaller than 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Also, notice that the estimated number of groups increases with H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Visualizations of how adjusted rand indices and estimated number of groups changes over H are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' H MaxPEAR MinBinder GreedyEPL 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='4067 (3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='4008 (5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='4321 (3) 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='4882 (3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='4977 (6 ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='6521 (4) 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='7374 (5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='7115 (11) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='6635 (4) 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='7237 (6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='7442 (20) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='7134 (4) 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='7449 (7) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='6624 (25) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='7313 (4) 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='7422 (8) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='6674 (25) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='7293 (8) 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='7056 (12) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='7041 (25) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='7043 (11) Table 9: Adjusted Rand indices of different estimates under LPJMM with different H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Numbers in the parentheses denote the numbers of estimated groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 3 4 5 6 7 8 9 H Adjusted rand index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='8 3 4 5 6 7 8 9 H number of groups 5 10 15 20 25 MaxPEAR MinBinder GreedyEPL Figure 9: Left: Adjusted rand indices of the clustering estimates found by using the MaxPear, MinBinder, and GreedyEPL methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' Right: Estimated number of groups using the three methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' The goodness-of-fit test outlined in Section 3 is also carried out here to compare the means of several sum- mary statistics, which are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' As we can see from the plots, the model’s fit is not affected by the choice of H even for H smaller than the actual number of clusters in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 20 3 4 5 6 7 8 9 H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='1536 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='1540 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='1544 (a) density 3 4 5 6 7 8 9 H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='513 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='514 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='515 (b) transitivity 3 4 5 6 7 8 9 H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='538 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='546 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='554 (c) assortativity Figure 10: The means of summary statistics for different H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' C Traceplots of log-likelihood The traceplots of the log-likelihood (after thinning the Markov chain every 10 iterations) in simulation stud- ies and real applications in Sections 3 and 4 are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 0 2000 6000 10000 −1100 −1080 −1060 −1040 (a) simulation 1 0 2000 6000 10000 −1920 −1900 −1880 −1860 (b) simulation 2 0 2000 6000 10000 −5230 −5214 −5198 −5182 (c) Lazega network Figure 11: Traceplots of the log-likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' D Visualizations of results from LPJMM and LPCM Visualizations of the estimated latent positions and estimated group membership using the MaxPEAR, Min- Binder, and GreedyEPL methods under LPJMM and LPCM are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' 12 and 13 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='5 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='99 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='(c) GreedyEPL (LPCM) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content='Figure 13: Points are plotted based on estimated z and three estimated ˆg of LPCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' References J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'} +page_content=' H.' metadata={'source': 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