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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf,len=5504
+page_content='Hyper-cores promote localization and efficient seeding in higher-order processes  Marco Mancastroppa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='1 Iacopo Iacopini,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='2 Giovanni Petri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='3 and Alain Barrat1  1Aix Marseille Univ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Universit´e de Toulon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' CPT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Turing Center for Living Systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Marseille,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' France  2Department of Network and Data Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Central European University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' 1100 Vienna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Austria  3CENTAI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Corso Inghilterra 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' 10138 Turin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Italy  Going beyond networks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' in order to include higher-order interactions involving groups of elements  of arbitrary sizes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' has been recognized as a major step in reaching a better description of many  complex systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In the resulting hypergraph representation, tools to identify particularly cohesive  structures and central nodes are still scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' We propose here to decompose a hypergraph in hyper-  cores, defined as subsets of nodes connected by at least a certain number of hyperedges (groups  of nodes) of at least a certain size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' We illustrate this procedure on empirical data sets described  by hypergraphs, showing how this suggests a novel notion of centrality for nodes in hypergraphs,  the hyper-coreness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' We assess the role of the hyper-cores and of nodes with large hyper-coreness  values in several dynamical processes based on higher-order interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' We show that such nodes  have large spreading power and that spreading processes are localized in hyper-cores with large  connectedness along groups of large sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In the emergence of social conventions moreover, very  few committed individuals with high hyper-coreness can rapidly overturn a majority convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Our work opens multiple research avenues, from fingerprinting and comparing empirical data sets  to model validation and study of temporally varying hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  INTRODUCTION  Network theory provides a powerful framework for de-  scribing a wide range of complex systems composed of  elements interacting in pairs [1–4]: over the years, this  theory has developed numerous concepts and techniques  to characterize the structure of complex networks at var-  ious scales, from the single element (node or link) to  groups of nodes to the whole system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Moreover, net-  works are the support of dynamical processes of various  types, from spreading to synchronization phenomena [3],  thus understanding how network’s features impact such  processes, or which parts of a network play the most im-  portant role, is of primordial relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Several concepts  and results in this respect are now well established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' For  instance, hubs, nodes with a very large number of con-  nections (degree), are known to influence processes such  as spreading or opinion dynamics, because of their ten-  dency to be reached easily and their ability to transmit to  many other nodes [1, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The statistics of the individual  number of connections of nodes are however not a rich  enough characterization: the existence of well-connected  groups of nodes might indeed be even more relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In  this direction, the tendency of hubs –observed in real-  world networks– to be connected to each other far above  chance is quantified by the rich-club coefficient [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  A  more systematic way to decompose a network into a hi-  erarchy of subgraphs of increasing connectedness is given  instead by the k-core decomposition [6–9]: the k-core of  a network is by definition the maximal subgraph such  that all its nodes have degree (number of neighbours in  the subgraph) at least k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' This decomposition provides a  fingerprint of the network’s structure [8, 10, 11] and grad-  ually focuses on more central and densely interconnected  parts of the network that were shown to play a crucial  role in spreading processes [12–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In fact, the coreness  of a node, defined as the largest value of k such that the  node belongs to the corresponding k-core, gives an alter-  native measure of centrality and largely determines the  impact of a spreading process initiated (seeded) in that  node [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Given the relevance of this decomposition, it  has also been extended to weighted networks [15], via the  s-core decomposition [16] (s representing the strength of  a node, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=', the sum of the weights of its adjacent links),  to temporally evolving networks [17, 18], and to multi-  layer networks [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Despite their convenience, network representations are  limited to systems composed of only binary interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  However, over the last few years, it has become clear that  many real systems include interactions between groups  of units [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Examples range from group conver-  sations among friends [22] to research teams and co-  authorship of scientific articles [23], from neural systems  [24] to interactions between multiple species in ecologi-  cal ones [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Analogously, considering a purely dyadic  network substrate for the unfolding of processes such  as consensus formation or (social) contagion could put  a limit on the ability to describe key mechanisms that  are at play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' For instance, reinforcement mechanisms –  in which two or more people can convince others in a  group conversation– cannot be naturally accounted for  when considering only dyadic interactions [26–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  In  these cases, systems and processes can be effectively rep-  resented within the framework of hypergraphs, a “higher-  order” generalization of networks in which nodes can in-  teract in hyperedges, groups of arbitrary size [21, 30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Higher-order interactions give rise to novel structures [32–  34] and phenomena [20, 35], highlighting the need for new  characterization tools able to detect hierarchies and rele-  vant subparts of all these systems that are better repre-  sented by hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Here, we contribute to this endeavour by proposing  the decomposition of a hypergraph in (k, m)-hyper-cores,  which we define as a series of subhypergraphs of increas-  ing connectivity k, ensured by hyperedges of increasing  sizes m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' We apply this decomposition to a wide range of  data sets, representing systems of different nature, iden-  tifying non-trivial mesoscopic higher-order structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In  so doing, we put forward the hyper-coreness, a new cen-  trality measure for nodes in hypergraphs based on their  inclusion in the hyper-cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Finally, and crucially, we  investigate the role of the newly defined hyper-cores  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='04235v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='soc-ph]  10 Jan 2023  2  and of the nodes with largest hyper-coreness in spread-  ing and consensus processes based on group interactions  [28, 36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' We show that spreading processes tend to  be localized on hyper-cores associated to large k and  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' We then study the performance of hyper-coreness-  based strategies as opposed to both random and strength-  based ones [16] when it comes to identifying influential  nodes that sustain and drive higher-order processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' We  find that hyper-coreness can be effectively used to max-  imise the total outbreak size in non-linear spreading pro-  cesses [36] and help committed minorities reach the tip-  ping point leading to a systemic takeover in social con-  vention games [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  RESULTS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Hyper-core decomposition and hyper-coreness  We define the hyper-cores, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' the higher-order cores  of a hypergraph, through a systematic decomposition  of a hypergraph in a double hierarchy of nested sub-  hypergraphs of increasing connectedness and hyperedge  sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Let us consider a (static) hypergraph H = (V, E),  where V is the set of its N = |V| nodes and E is the  set of its hyperedges [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  We recall that a hyperedge  e = {i1, i2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=', im} is a set of m nodes, which can thus  represent a group interaction between these nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' We  denote by M = maxe∈E |e| the largest hyperedge size in  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Each node i ∈ V can be characterized by a vector  of degrees d(i) = [d2(i), d3(i), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=', dm(i), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=', dM(i)] whose  component dm(i) denotes the m-hyper-degree of the node  i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=', the number of distinct hyperedges of size m to  which it belongs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  We denote by Dm(i) = �  p≥m dp(i)  the number of distinct hyperedges of size at least m to  which i belongs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  We define the (k, m)-hyper-core as the maximum sub-  hypergraph J induced by the set of nodes A ⊆ V and  with hyperedges of size at least m, such that ∀ i ∈  A, DJ  m(i) ≥ k, where DJ  m(i) denotes the number of dis-  tinct hyperedges of size at least m in which i is involved  within the subhypergraph J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In other terms, all the nodes  in the (k, m)-hyper-core belong to at least k hyperedges  of size at least m, within the hyper-core itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The set of  hyperedges of the subhypergraph J , induced by the set  A ⊆ V, is defined by S = {e ∩ A|e ∈ E ∧ |e ∩ A| ≥ m}  [39] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=', a hyperedge of S is a subset of a hyperedge of  E, of size at least m and containing only nodes of A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  To obtain the (k, m)-hyper-core of a hypergraph, one  can first remove from E all hyperedges of size smaller than  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' One then removes recursively from V all nodes i with  Dm(i) < k, until all the nodes in the remaining subhy-  pergraph are involved in at least k hyperedges of size at  least m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Note that this process does not correspond only  to the removal of nodes with Dm(i) < k in the original  hypergraph H: indeed, each time a node is removed, the  sizes of the hyperedges to which it belongs decrease by  one unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Thus, the removal of a node can induce the  removal of some of the hyperedges to which it belongs, if  their size becomes less than m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' 1 we illustrate the  process on an example hypergraph and highlight some of  its (k, m)-hyper-cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  As k and m increases, the (k, m)-hyper-cores progres-  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Sketch of the (k, m)-hyper-core decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  We show a hypergraph and highlight some of its (k, m)-hyper-  cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Note the inclusions as k or m increase: the (1, 2)-  hyper-core contains the (1, 3)-hyper-core, which contains the  (2, 3)-hyper-core;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' similarly the (1, 2)-hyper-core contains the  (2, 2)-hyper-core which contains the (2, 3)-hyper-core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' On the  other hand, the (1, 3)-hyper-core and the (2, 2)-hyper-core  share some nodes but neither is included in the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The  green nodes belong to the (1, 2)-hyper-core but neither to the  (1, 3)- nor the (2, 2)- ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  The blue nodes belong to the  (1, 3)-hyper-core but are excluded from the (2, 3) one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Or-  ange nodes belong to the (2, 2)-hyper-core but are excluded  from the (2, 3) one because they belong only to hyperedges  of size 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The (1, 4)-core and (1, 5)-core contain all the nodes  involved respectively in at least one interaction with m ≥ 4  and m ≥ 5 (for simplicity these cores are not highlighted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  The (k, 2)-cores and (k, 3)-cores with k ≥ 3, and the (k, 4)-  cores and (k, 5)-cores with k ≥ 2 are all empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Notice that  the node i does not belong to the (2, 3)-core even if D3(i) = 2  because of the recursive and interaction downgrading mecha-  nisms of the decomposition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' in the (1, 3)-core and (2, 3)-core  the pairwise interactions ei ∀i ∈ [1, 5] are excluded, thus the  (1, 3)-core is composed of two disjoint subhypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  sively identify groups of nodes increasingly connected  with each other through interactions of increasing or-  der (the (k, m)-hyper-core includes the (k, m + 1)- and  (k + 1, m)-hyper-cores).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  We define the m-shell index  Cm(i) of a node i as the value of k such that i belongs  to the (k, m)-hyper-core but not to the (k + 1, m)-hyper-  core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The (k, m)-shell S(k,m) can then be defined as the  set of all nodes whose shell index Cm(i) at size m is k,  and we denote by km  max the maximum value of k such  that the shell S(k,m) is not empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The ratio Cm(i)/km  max  thus quantifies how well-connected node i is in the hy-  pergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' As it is a function of m, different nodes will  have different functions with potentially different func-  tional shapes, which makes it difficult to compare and  rank them (see the Supplemental Material, SM, for some  examples of Cm(i) functional shapes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' We thus define for  each node i its hyper-coreness R(i) as:  R(i) =  M  �  m=2  g(m)Cm(i)/km  max,  (1)  where g(m) is an arbitrary weight function, which can  3  weigh differently the various possible sizes of higher-order  interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Hereafter, for simplicity we will fix g(m) =  1: in this case, by definition R ∈ [0, M −1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Other choices  could be considered, for example to emphasise hyperedges  of larger or smaller sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The R hyper-coreness gives thus  a summary of a node centrality with respect to the various  (k, m)-hyper-cores, taking into account how central the  node is for all orders of interactions and making it possible  to rank the nodes of the hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Hyper-core decomposition of empirical  hypergraphs  To illustrate the decomposition processes along (k, m)-  hyper-cores, we rely on a number of empirical hyper-  graphs, obtained from publicly available data sets, that  describe a variety of systems of agents interacting in dif-  ferent environments, both through online media and face-  to-face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In particular, we consider data sets of face-to-face  interactions provided by the SocioPatterns collaboration  [40–42] and by the Contacts among Utah’s School-age  Population (CUSP) project [43], collected in contexts  ranging from workplaces to schools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  We also use data  sets of email communication (email-EU, email-Enron [44–  46]) and of other types of online interactions, namely on-  line reviews of products (music-review [46, 47]) or online  opinion exchange on specific topics in scientific forums  [46, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' We moreover consider data describing commit-  tees membership (house-committees, senate-committees  [46, 49, 50]) and bills sponsorship (congress-bills, senate-  bills [46, 49, 51, 52]) in the US Congress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  These data  sets cover a wide range of system sizes and have also  very different interaction size distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' We provide  a detailed description of each data set in the Methods  and in the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In the following, we give results on the  music-review, email-EU, house-committees and congress-  bills data sets while we refer to the SM for the other data  sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Figure 2 shows the results of the hyper-core decompo-  sition on two data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The relative size n(k,m) of the  (k, m)-hyper-cores exhibit distinct behaviors as a func-  tion of k and m, identifying structural differences between  data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In some cases, the decrease with k is rather  smooth (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' 2a and SM), showing that most shells are  populated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In other cases abrupt drops and plateaus can  be observed (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' 2c and SM), corresponding to alter-  natively empty and densely populated (k, m)-shells (see  also SM for figures showing the sizes of the (k, m)-shells  vs k and m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' These differences indicate that the (k, m)-  hyper-cores could be used to provide a fingerprint of hy-  pergraphs, just as the k-core decomposition provides a  fingerprint of networks [8, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Also the distributions  of hyper-coreness values R differ across data sets, as illus-  trated in the rank-order plots of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' 2b,d and in the SM  for all data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' While some data sets have an almost  uniform distribution of values, others feature few nodes  with high hyper-coreness and many nodes with medium  hyper-coreness, or vice-versa –many nodes having low or  high coreness and few with medium values (see SM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' We  also show in the SM some typical examples of the nor-  malized m-shell index function Cm(i) as a function of m  for various nodes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' to highlight the diversity of these func-  1  16 31 46 61 76 91 106  k  2  4  6  8  10  12  14  16  18  20  22  24  m  n(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' m)  a  email-EU  0  250  500  750  1000  Rank(i)  0  5  10  15  20  R(i)  b  1  6 11 16 21 26 31 36  k  2  12  22  32  42  52  62  72  82  m  n(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' m)  c  music-review  0  250  500  750  1000  Rank(i)  0  20  40  60  80  R(i)  d  10  1  100  0  50  100  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  n(k, m)  m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  0  500  1000  S(i)  0  10  20  R(i)  10  1  100  0  20  40  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  n(k, m)  m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  0  200  S(i)  0  40  80  R(i)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Hyper-core decomposition of empirical hyper-  graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Panels a and c show colormaps giving the relative  size n(k,m) (number of nodes in the hyper-core, divided by the  total number of nodes N) of the (k, m)-hyper-core as a func-  tion of k and m (white regions correspond to n(k,m) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In  the insets n(k,m) is shown as a function of k at fixed values of  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In panels b and d the hyper-coreness R(i) is plotted as a  function of the corresponding node rank;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' the insets give scat-  terplots of the hyper-coreness R(i) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' the s-coreness, S(i),  for all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In panels a and b we consider the email-EU  data set: R(i) and S(i) have a Pearson correlation coefficient  of ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='90 (p-value p ≪ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='001) and the corresponding rank-  ings have a Kendall’s τ coefficient of τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='85 (p ≪ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' in  panels c and d we consider the music-review data set: R(i)  and S(i) have a Pearson correlation coefficient of ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='74  (p ≪ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='001) and the corresponding rankings a Kendall’s τ  coefficient of τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='58 (p ≪ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  tions and the need to define a summary index such as the  hyper-coreness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  We finally compare in the insets of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' 2b,d the hyper-  coreness with the centrality of nodes obtained by disre-  garding the higher-order character of the interactions and  projecting the hypergraph H onto a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  To this  aim, we transform each hyperedge in a network clique,  and each edge (i, j) of the resulting network is weighted  by the number of distinct hyperedges in H involving both  i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' We then perform the s-core decomposition of this  weighted network and assign its s-coreness S(i) to each  node i [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' S(i) and R(i) are positively correlated, but  they do not provide exactly the same information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  In  particular the hyper-coreness enhances the information  given by the s-coreness by providing an internal hierar-  chy within the nodes of maximal s-coreness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' This is evi-  dent from the scatter plots, as nodes presenting the same  s-coreness values correspond to values of hyper-coreness  that can span across a broad range (y axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Having illustrated the relevance of the newly defined  cores on empirical hypergraphs, we now move to study  the role of these substructures in dynamical processes  taking place on hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In particular, we are go-  4  ing to investigate whether the (k, m)-hyper-cores and the  hyper-coreness centrality can be used to identify nodes  and structures relevant for spreading and consensus pro-  cesses whose mechanisms are explicitly defined on hyper-  edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Higher-order contagion processes localize in  hyper-cores  Networks have been widely used to describe the sub-  strate on which contagion processes take place, such as  the spread of a pathogen or information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  In standard  diffusion modeling approaches, nodes represent individ-  uals that at any time can be in one of several possible  states, such as S (susceptible), I (infectious) or R (re-  covered);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' S nodes are typically thought to become I at  rate β when they share a link with an infectious (I) in-  dividual, while infected (I) nodes recover spontaneously  at rate µ, either becoming again susceptible (S), in what  is usually called the SIS model [53], or becoming recov-  ered (R) in the so-called SIR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Recently, several  models have been proposed to take into account possible  higher-order mechanisms, that amount to reinforcement  mechanisms affecting the contagion probability due to the  simultaneous exposure to multiple sources of infections in  group interactions [26, 36, 54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' For instance, in a so-  cial contagion process, the probability that an individual  is convinced upon separate exposures to two “infectious”  neighbours can be reinforced if these exposures occur dur-  ing a group discussion featuring the three individuals al-  together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Here, we show that hyper-cores play a crucial role in  the dynamics of higher-order spreading processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In or-  der to do this, we consider the recently proposed higher-  order non-linear contagion [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In this model, each sus-  ceptible node in a hyperedge of size m in which there  are i infected individuals becomes infectious with rate  λiν, where ν controls the non-linearity of the process (for  ν = 1 the usual linear contagion is recovered, while for  ν > 1 non-linearities are introduced) and λ ∈ [0, 1] (see  Methods for more details on the model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Infected in-  dividuals (I) recover independently at constant rate µ,  becoming either susceptible S (SIS model) or R (SIR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  The higher-order nature of contagion produces novel  effects on the epidemic phenomenology, including abrupt  transitions with bistability in the SIS phase diagram, in-  termittent regimes [37, 54], and a mesoscopic localization  of the infection on large hyperedges [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Against this  background, the connectivity properties of hyper-cores  we highlighted so far suggest that cores might play an  even stronger role in such localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  To investigate this point, we perform numerical simula-  tions of the higher-order non-linear SIS model on empiri-  cal hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The system is initialized with one single  seed of infection (a randomly chosen node in state I) in an  otherwise fully susceptible population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' We let the process  evolve (see Methods for simulation details) until a steady  state is reached in which the number of infectious individ-  uals fluctuates (we consider parameter values such that  the system remains active and the epidemic does not die  out rapidly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' We then consider a finite time-window T  and measure for each node j the time τ(j) that it spends  1  6 11 16 21 26 31 36  k  2  12  22  32  42  52  62  72  82  m  /T  a  music-review  SIS  1  6 11 16 21 26 31 36  k  2  12  22  32  42  52  62  72  82  m  R  b  SIR  1 3 5 7 9 1113151719  k  2  12  22  32  42  52  62  72  82  m  /T  c  house-committees  1 3 5 7 9 1113151719  k  2  12  22  32  42  52  62  72  82  m  R  d  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='85  0  500  1000  n  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='9  /T n  S-rank  R-rank  500  600  700  800  900  0  1000  n  400  600  800  R  n  S-rank  R-rank  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='675  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='700  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='725  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='750  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='775  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='800  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='825  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='850  0  500 1000  n  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='9  /T n  S-rank  R-rank  100  125  150  175  200  225  0  500 1000  n  100  200  300  R  n  S-rank  R-rank  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Hyper-cores for seeding and localization in  higher-order non-linear contagion processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' For the  SIS model, panels a and c give the heatmap of the average  fraction of time ⟨τ/T⟩ of infected nodes in the steady state as  a function of k and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Averages are computed over all the  nodes of each (k, m)-hyper-core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The insets represent ⟨τ/T⟩  averaged over the first n nodes according to the coreness rank-  ings as a function of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' All results are obtained by averaging  the results of 103 numerical simulations, with an observation  window T = 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' For the SIR model, panels b and d show  the heatmap of the average final size of the epidemic ⟨R∞⟩  as a function of k and m, where the process is seeded in a  single node belonging to the (k, m)-hyper-core (averaged over  all nodes of the hyper-core).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The insets represent ⟨R∞⟩ as  a function of n averaged over the first n nodes according to  coreness rankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' All results are obtained by averaging the  results of 300 numerical simulations for each seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Panels a  and b: music-review data set with ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='25, λ = 5 × 10−4  (a) and ν = 3, λ = 5 × 10−4 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Panels c and d: house-  committees data set with ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='25, λ = 5 × 10−4 (c) and  ν = 4, λ = 5 × 10−5 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In all panels µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  in the I state during that window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' This allows to iden-  tify the nodes on which the epidemic is mainly localized  in the steady state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' the nodes that drive and sustain  the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Figure 3 reports results of simulations performed on  two data sets (the music-review and house-committees  data sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Similar results are shown in the SM for the  other considered data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Panels 3a and 3c show that  nodes in (k, m)-hyper-cores with either increasing k or m  tend to be more often infectious, as the values of τ(j)/T  averaged over all nodes of each (k, m)-hyper-core increase  with k and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' This implies that the process is more local-  ized in the (k, m)-hyper-cores with large k (which favors  connectedness, hence mutual reachability) and m (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=',  large hyperedges where large values of i can be obtained  yielding large infection rates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  The insets of the pan-  els moreover show the average of τ/T over the n nodes  with highest hyper-coreness R or highest s-coreness S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  The nodes with highest coreness tend to be more often in  the infectious state, and this tendency is stronger for the  5  hyper-coreness than for the s-coreness: among the nodes  with largest value of s-coreness, the hyper-coreness al-  lows to distinguish which ones are most involved in the  higher-order spreading processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Moreover, in the SM  we show that a similar phenomenology is obtained with a  different model of contagion involving higher-order mech-  anisms [37, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  High hyper-coreness seeds increase total  outbreak size  Nodes belonging to large interaction groups have also  been shown to be optimal seeds of higher-order non-linear  contagions in terms of spreading speed at the beginning  of an SIS outbreak [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Which nodes have the largest  spreading power in the long run, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=', in terms of final  size reached by a SIR process [12], remains however an  open question for higher-order spreading processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' We  thus consider the higher-order non-linear SIR model, in  which the dynamics, starting from a single seed, evolves  until no individual is in the state I anymore (only nodes  in states S or R remain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  To quantify the “spreading  power” of each node j considered separately as seed, we  average the final epidemic size R∞(j), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=', the number  of nodes in state R at the end of the process, over 300  stochastic runs for each seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Figure 3b and 3d show  that this average final epidemic size ⟨R∞(j)⟩, averaged  over all nodes of each (k, m)-hyper-core, increase with k  and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The insets also show that the nodes with higher  hyper-coreness lead to larger epidemics, determining a  hierarchy even among the nodes with highest s-coreness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  In summary, nodes with higher connectedness along  groups of larger sizes can seed more efficiently, and the  hyper-coreness provides a good identification of the nodes  with highest spreading power in higher-order non-linear  contagion processes (similar results are shown in the SM  for another higher-order contagion model [37, 54]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Hypercore seeding facilitates systemic takeover  by minority norms  Group interactions can also play an important role in  the formation of a consensus and the emergence of shared  conventions in a population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' According to critical mass  theory, regular individuals might then benefit –towards  addressing societal challenges– from the presence of a  committed minority that aims at overturning the sta-  tus quo [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Recently, it has been shown that groups  can modulate this takeover [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' An important issue in  this respect concerns the best “seeding” strategy –where  should the committed minority start from in order to best  achieve the takeover?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Here we show how hyper-coreness  can provide an answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  We consider the well-known naming-game (NG) model  [38], which describes how a shared convention can emerge  in a population of locally interacting agents [57, 58], and  has been shown to account for the outcome of controlled  experiments of social coordination [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In the minimal  version of this model, recently modified to take group in-  teractions into account [28], individuals are represented  by the N nodes of a static hypergraph, and each node is  endowed with a dictionary that can contain at most two  names (representing conventions or norms), A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  At each time-step a hyperedge is chosen randomly and  a speaker is randomly selected within it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  The speaker  randomly chooses a name from its dictionary and com-  municates it to the other hyperedge members (the listen-  ers), who can agree or not on the proposed name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' To  determine the possibility of an agreement within the hy-  peredge, we consider two distinct alternatives [28]: (i)  the union rule, for which an agreement can be reached if  at least one of the listeners has the proposed name in its  dictionary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' (ii) the unanimity rule, for which the agree-  ment can be reached only if all the nodes in the group  have the proposed name in their dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' A parame-  ter β ∈ [0, 1] modulates the social influence by controlling  the propensity of the listeners to actually accept the local  consensus: the agreement in the group becomes effective  only with probability β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In this case, all nodes in the hy-  peredge add the accepted name to their dictionary, if it  was not already present, deleting all others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' If instead no  agreement is reached, the listeners simply add the name  given by the speaker to their dictionaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Crucially, we include in the population a committed  minority of Np individuals who do not obey the aforemen-  tioned rules whenever they are listeners, but they instead  stick to their norm, a single name A, and their dictionary  is never updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' We initiate the process with the rest of  the population, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' the majority, having only the name  B in their dictionaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The system can evolve towards  different regimes of co-existence of the two names or of  dominance of one name over the other, depending on β,  on the considered rule, and on the relative size of the mi-  nority p = Np/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In particular, the committed minority  can overcome the majority, with the whole population  eventually converging on A, for a range of intermediate  values of β and for large enough p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  When committed  individuals are chosen at random in the population, this  range increases when the hypergraph contains hyperedges  of larger sizes [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' This naturally raises the question of  whether the committed minority might also benefit from  belonging to specific substructures of a given hypergraph,  such as hyper-cores with large connectedness and group  sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Here we investigate this issue through numerical simu-  lations of the higher-order NG process on empirical static  hypergraphs for varying values of the parameter β and of  the fraction p of committed individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In our numeri-  cal experiments committed individuals are selected with  different seeding strategies: (i) at random from the entire  population (random);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' (ii) as the Np ones with the high-  est hyper-coreness R (top hyper-coreness);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' (iii) as the Np  ones with the highest s-coreness (top s-coreness) in the  projected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In each experiment we measure the frac-  tion nA of nodes holding only A in their dictionary (both  committed or not), and focus on its large time limit n∗  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  This limit can be either 1, if the population reaches the  absorbing state in which all nodes agree on A, or, if the  absorbing state is not reached before tmax time-steps, we  average over 100 values of nA(t) sampled from the last T  time-steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Figure 4 reports the simulation results for two empir-  ical data sets, congress-bills (a-d) and the email-EU (e-  h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The results for the other data sets can be found in  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  nA(t)  h  Random  R-rank  s-rank  1/N  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  1/N  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Comparison of seeding strategies for committed minorities in a higher-order naming-game process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In  the heatmaps a-c, e-g the stationary fraction n∗  A of nodes supporting only the name A is shown as a function of the fraction  of committed nodes p and the agreement probability β (steps correspond to the fact that p varies by increments of 1/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' a-c:  congress-bills data set with unanimity rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' e-g: email-EU data set with union rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Committed nodes are selected through  different strategies, in particular random seeding (a,e), top s-coreness (b,f), and top hyper-coreness (c,g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Panels d,h show  the temporal evolution of nA(t) according to the different seeding strategies and for fixed values of β and p (cross markers in  the heatmaps), (d): (β, p) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='48, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='3 × 10−2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' (h): (β, p) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='55, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='2 × 10−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The minority takeover, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' n∗  A = 1, takes  place for 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='9% of the explored parameter space in panel a, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='3% in b, 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5% in c, 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0% in e, 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='9% in f, and 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='4% in g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' All  simulations are run until the absorbing state n∗  A = 1 is reached or the dynamics has evolved for tmax = 5 × 105 time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The  stationary fraction n∗  A is obtained by averaging over 100 values sampled in the last T = 5 × 104 time-steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Results refer to the  median values obtained over 200 simulations for each pair of parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' For the random strategy, we recover the results  of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' [28]: for low values of β, a co-existence state of  A and B is observed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' at low fraction of committed and  large β values , the majority remains B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Contrarily, at  intermediate values of β, the minority takes over and the  whole population converges on A (in a way favored by the  union rule w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' the unanimity rule).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Non-random seed-  ing strategies yield the same phenomenology but enhance  the range of parameters in which the minority overturns  the majority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' This is especially the case when we place  committed individuals in the most central nodes accord-  ing to their hyper-coreness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In this case, a tiny fraction  of committed is able to take over on a wide range of β  values, while for lower β values, a co-existence regime  is always observed –due to the small propensity to ac-  cept a local consensus [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' For example, with the top  hyper-coreness strategy a fraction p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='51 × 10−2 in  the congress-bills data set with unanimity rule is enough  to allow the minority takeover over a range of β values  whose extension is ∆β ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' This cannot be achieved  with the other two seeding strategies, for which below  p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='8 × 10−2 only ∆β ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='2 can be reached (see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' 4a-c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Analogously, in the email-EU data set with  the union rule a fraction p = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='1 × 10−3 is enough to ob-  tain the minority dominance over ∆β ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5 when seeded  according the top hyper-coreness strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In this case,  with the top s-coreness and the random strategies the  same result is obtained only for p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='33 × 10−2 and  p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='74 × 10−2 respectively (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' 4e-g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' To further  quantify the differences among these strategies we can  also calculate the value of critical mass pc necessary to  bring the system to the tipping point while keeping β con-  stant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In the congress-bills data set with unanimity rule  and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='62 for instance, the critical mass for the top  hyper-coreness strategy is pc = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='4 × 10−3 as compared  to pc = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='68 × 10−2 and pc = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='04 × 10−2 obtained with  the random and the top s-coreness strategies respectively  (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' 4a-c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' similarly, in the email-EU data set with  union rule and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='83, these values are respectively  pc = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='1 × 10−3, pc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='53 × 10−2, pc = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='2 × 10−3 (see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' 4e-g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  The hyper-coreness centrality is thus particularly effec-  tive in identifying nodes with a crucial role in higher-order  NG processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Indeed, nodes belonging to (k, m)-hyper-  cores with large values of k and m, if committed, can  convince many others through their simultaneous pres-  ence in several large groups, and this can be efficiently  sustained by their large connectedness, favouring conver-  gence on their convention even outside the committed  minority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In addition, the temporal evolution diplayed in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' 4d,h illustrate how, even when all seeding strategies  lead to the agreement on the convention initially sup-  7  ported by the minority, the convergence is much faster  for the hyper-coreness seeding strategy, followed by the  s-coreness and the random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  DISCUSSION  We have considered here a systematic procedure to ex-  tract, from a given hypergraph, structures of increasing  connectedness along increasing group sizes: the (k, m)-  hyper-cores, in which each node is connected to the other  by at least k hyperedges (representing higher-order in-  teractions) of sizes at least m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Using the maximal con-  nectedness values of each node we define a new concept of  centrality in hypergraphs: a node hyper-coreness summa-  rizes its relative depth in the hierarchies of hyper-cores at  all orders (interaction sizes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Applying these concepts to  empirical data describing a variety of higher-order sys-  tems, we have shown how the (k, m)-hyper-cores pro-  vide a fingerprint of empirical hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Crucially,  we have also highlighted how hyper-cores with increas-  ing k and m play important roles in several dynamic  processes with higher-order mechanisms unfolding upon  hypergraphs, such as contagion processes and consensus  formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  The hyper-coreness centrality in particular  identifies nodes with high spreading power and on which  stationary contagion processes tend to localize;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' moreover  nodes with high hyper-coreness, if belonging to a commit-  ted minority, can be particularly efficient at overturning  a majority convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Our work opens the door to several research directions  in the expanding field of hypergraphs structure and dy-  namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' It can provide an additional systematic charac-  terization of both empirical and model hypergraphs, and  thus potentially a model validation tool as well as a com-  parison method between hypergraphs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' by computing  distances between the (k, m)-hypercore profiles of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  For specific systems where additional properties of the  nodes are known, the shell indices and hyper-coreness  values of nodes could be compared in more detail to pro-  vide insights into their relative positions and roles in the  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Moreover, while here we focused on static hypergraphs,  many such systems evolve in time [60, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Hyper-cores  and hyper-coreness could be used to investigate the evo-  lution of the higher-order interactions at multiple scales,  from the global evolution of the structure described by  hyper-core sizes, to the shell indices and hyper-coreness  of individual nodes from one period to the next [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' An  interesting case study in this direction could be for in-  stance the evolution of the hyper-core positions of scien-  tists in co-authorship “networks”, which are in fact by  construction evolving hypergraphs [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  MATERIALS AND METHODS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Data description and preprocessing  Several data sets we considered are publicly available in  the form of static hypergraphs, thus they do not require  any preprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' These data sets describe:  email communications: within a European institu-  tion (email-EU [44]), and within Enron, between  a core-set of workers (email-Enron [45, 46]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Each  node corresponds to an email address and a hy-  peredge includes the sender and all receivers of an  email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  interactions in legislative bills in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Congress  (congress-bills) and in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Senate (senate-bills)  [46, 49, 51, 52]: each node corresponds to a member  of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Congress or Senate and a hyperedge  involves sponsors and co-sponsors of legislative bills  discussed in the Congress or Senate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  interactions in committees in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' House of  Representatives (house-committees) and in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Senate (senate-committees) [46, 49, 50]: each node  corresponds to a member of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' House of Rep-  resentatives or Senate and each hyperedge involves  nodes that share membership in a committee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  online interactions (3 data sets):  exchanges be-  tween users of MathOverflow on algebra top-  ics  (algebra-questions)  or  on  geometry  topics  (geometry-questions), in which each node corre-  sponds to a user of MathOverflow and each hyper-  edge involves those users who have answered a spe-  cific question belonging to the topic of algebra or ge-  ometry [46, 48];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' interactions between Amazon users  on music (music-review [46, 47]), in which each node  corresponds to an Amazon user and each hyperedge  involves users who have reviewed a specific product  belonging to the category of blues music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Moreover, we built static hypergraphs from several  data sets of time-resolved face-to-face human interac-  tions, as in [26, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The data sets are provided by the  SocioPatterns collaboration [40–42] and by the Contacts  among Utah’s School-age Population (CUSP) project  [43] and describe interactions between individuals in sev-  eral contexts :  a hospital (LH10 [62]), a workplace  (InVS15 [41, 63]), a conference (SFHH [41]), a high-school  (Thiers13 [64]), two primary-schools (LyonSchool [65],  Elem1 [43]) and a middle-school (Mid1 [43]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' For these  data sets we carried out an aggregation procedure to ob-  tain static hypergraphs: (i) we aggregate the data over  time windows of 15 minutes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' (ii) we identify the cliques  in each time window, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' groups of nodes forming a fully  connected cluster, (iii) we identify in each temporal win-  dow the maximum cliques, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  cliques not completely  contained in a larger clique, and promote them to a hy-  peredge status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Overall, the data sets considered describe interactions  in several different environments, mediated by different  mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' They correspond to a wide variety of sta-  tistical properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' data set size, hyperedges size dis-  tributions), as shown in the SM where these statistical  properties of the data sets are reported in details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  8  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Models and stochastic simulations  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Higher-order non-linear contagion  We performed stochastic numerical simulations of the  higher-order non-linear contagion model on each empir-  ical static hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  The simulations are performed  with discrete time-steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The S → I infection mecha-  nism is the same for the SIR and the SIS models: for  each time-step ∆t, given a hyperedge of size m contain-  ing i infected nodes, each of the susceptible nodes in it  can be infected with probability (1 − e−λiν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Therefore,  the probability that a node j is infected in a time-step  ∆t is:  pj = 1 −  �  e∈E(j)  e−λiν  e ,  (2)  where E(j) denotes the set of hyperedges in which the  node j is involved and ie is the number of infected nodes  in the hyperedge e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Each infected node heals (return-  ing susceptible in SIS or gaining immunity in SIR) with  probability µ in each time-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  In the SIS process the population is initialized with a  single infectious seed randomly selected in the population  and the process is iterated until the system reaches a  steady state with a fluctuating number of infectious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' An  observation time window T is then considered and the  time τ spent in the infectious state is estimated for all  nodes over that time-window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The results are averaged  over 103 simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  In the SIR process the population is initialized with a  single infectious seed j and the dynamic process is iter-  ated until no more infectious nodes are present: the final  epidemic size R∞(j) obtained by seeding the infection in  j is defined as the final number of nodes in the R state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  The results are averaged over 300 simulations for each  infection seed j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Higher-order NG process  We also performed numerical simulations of the higher-  order NG process on the empirical hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  The  system with N nodes is initialized by fixing Np nodes as  belonging to the committed minority (equivalently, with  a fraction p = Np/N of committed nodes), with only the  name A in their dictionary, and setting the dictionaries  of all the other nodes of the majority with only the name  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The committed nodes are selected following one of  the three seeding strategies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' randomly from the whole  population or as the Np nodes with highest s-coreness or  hyper-coreness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' If several nodes have the same coreness  value, the committed nodes are randomly selected within  the coreness class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  The simulations are performed in discrete time-steps:  at each time-step a hyperedge is randomly selected (ac-  tivation of the group) and within it a node is randomly  chosen as the speaker, while the other nodes behave as  listeners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The speaker randomly selects a name in their  dictionary and all nodes in the group update their dictio-  nary according to the chosen agreement rule (except for  the committed nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The process is iterated until the  system reaches the absorbing state where all nodes have  only the name A in their dictionary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' nA(t) = n∗  A = 1,  or until the system has evolved for tmax time-steps: in  this last case the stationary fraction of nodes with the  name A in their dictionary n∗  A is obtained by averaging  nA(t) over 100 values sampled in the last T = 50, 000  time-steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The results refer to the median values ob-  tained over 200 simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' acknowledge support from the Agence  Nationale de la Recherche (ANR) project DATAREDUX  (ANR-19-CE46-0008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' acknowledges support from  the James S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' McDonnell Foundation 21st Century Sci-  ence Initiative Understanding Dynamic and Multi-scale  Systems - Postdoctoral Fellowship Award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  [1] Albert, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' & Barab´asi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Statistical mechanics of  complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' 74, 47 – 97 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  [2] Dorogovtsev, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' & Mendes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Evolution of net-  works: From biological nets to the Internet and WWW  (Oxford University Press, Oxford, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  [3] Barrat, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=', Barthelemy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' & Vespignani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Dynami-  cal processes on complex networks (Cambridge university  press, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  [4] Newman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Networks (Oxford university press, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  [5] Colizza, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=', Flammini, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=', Serrano, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' & Vespignani,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Detecting rich-club ordering in complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Na-  ture Physics 2, 110–115 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  [6] Seidman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Network structure and minimum degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Social Networks 5, 269–287 (1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  [7] Batagelj, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' & Zaverˇsnik, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Fast algorithms for de-  termining (generalized) core groups in social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Advances in Data Analysis and Classification 5, 129–145  (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  [8] Alvarez-Hamelin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=', Dall’Asta, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' CPT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Turing Center for Living Systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Marseille,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' France  2Department of Network and Data Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Central European University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' 1100 Vienna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content=' Corso Inghilterra 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content=' Italy  In this Supplementary Material we present the same results as in the main text for all the considered data sets  and also further results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' We first present in detail some of the statistical properties of the data sets and of the  static hypergraphs considered (Supplementary Section 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In Supplementary Section 2 we present the results of  the (k, m)-core decomposition, showing how the (k, m)-cores and (k, m)-shells are populated as a function of k  and m, the functional form of the m-shell index Cm(i) for some nodes, the distributions of the hypercoreness and  s-coreness centralities and their correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In Supplementary Section 3 we present the results of the higher-order  non-linear contagion process [1], both in the SIS and SIR formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In Supplementary Section 4 we introduce  in details the threshold higher-order process [2], its numerical implementation and its results in relation to the  hyper-cores, both in the SIS and SIR formulation, as done for the higher-order non-linear contagion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Finally  in Supplementary Section 5, the results of the higher-order naming-game process [3] are presented for both the  union and the unanimity rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Supplementary Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Properties of the data sets  The considered data sets describe interactions in several environments, mediated by different mechanisms, and  thus they differ in their fundamental statistical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' This is summarized in Table I and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' 1: the number  of nodes and hyperedges vary among the data sets considered, the distribution Ψ(m) of the hyperedge sizes m, the  range of their sizes m ∈ [2, M] and the average hyperedge size ⟨m⟩ are different among the data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  data set  N  E  M  ⟨m⟩  LH10  76  1 102  7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='Supplementary Figure 1: Hyperedge size distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content=' the  number of hyperedges of size m, for all the data sets: LH10 (panel a), Thiers13 (panel b), InVS15 (panel c),  SFHH (panel d), LyonSchool (panel e), Mid1 (panel f), Elem1 (panel g), email-EU (panel h), congress-bills  (panel i), senate-committees (panel j), email-Enron (panel k), house-committees (panel l), music-review (panel  m), senate-bills (panel n), algebra-questions (panel o) and geometry-questions (panel p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='Supplementary Figure 2: Hyper-core decomposition I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' All panels show colormaps giving the relative size  n(k,m) (number of nodes in the hyper-core, divided by the total number of nodes N) of the (k, m)-hyper-core as a  function of m and k (white regions correspond to n(k,m) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The following data sets are considered: LH10  (panel a), Thiers13 (panel b), InVS15 (panel c), SFHH (panel d), LyonSchool (panel e), Mid1 (panel f), Elem1  (panel g), email-EU (panel h), congress-bills (panel i), senate-committees (panel j), email-Enron (panel k),  house-committees (panel l), music-review (panel m), senate-bills (panel n), algebra-questions (panel o) and  geometry-questions (panel p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  4  0  20  40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  n(k, m)  a  m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  0  20  40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  b  m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  0  20  40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  c  m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='0  0  10  20  30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  d  m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='0  0  100  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  n(k, m)  e  m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  f  m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='0  h  m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='0  m = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  0  10  20  30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  j  m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  0  10  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  k  m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  5  10  15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  l  m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  0  20  40  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  n(k, m)  m  m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  0  500  1000  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  n  m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  0  20  40  60  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  o  m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  0  25  50  75  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  p  m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  m = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  Supplementary Figure 3: Hyper-core decomposition II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' All panels show the relative size n(k,m) (number of  nodes in the hyper-core, divided by the total number of nodes N) of the (k, m)-hyper-core as a function of k for  fixed values of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The following data sets are considered: LH10 (panel a), Thiers13 (panel b), InVS15 (panel c),  SFHH (panel d), LyonSchool (panel e), Mid1 (panel f), Elem1 (panel g), email-EU (panel h), congress-bills  (panel i), senate-committees (panel j), email-Enron (panel k), house-committees (panel l), music-review (panel  m), senate-bills (panel n), algebra-questions (panel o) and geometry-questions (panel p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  5  1  9  17  25  33  41  49  2  3  4  5  6  7  m  s(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' m)  a  10  1  1  8  15  22  29  36  2  3  4  5  6  7  s(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' m)  b  10  2  10  1  1  9  17  25  33  41  2  3  4  5  6  7  8  9  10  s(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' m)  c  10  2  10  1  1  7  13  19  25  31  2  3  4  5  6  7  8  9  10  s(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='Supplementary Figure 4: (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' m)-shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' All panels show colormaps giving the relative size s(k,m) (number of  nodes in the hyper-shell, divided by the total number of nodes N) of the (k, m)-shell as a function of m and k  (white regions correspond to s(k,m) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The following data sets are considered: LH10 (panel a), Thiers13 (panel  b), InVS15 (panel c), SFHH (panel d), LyonSchool (panel e), Mid1 (panel f), Elem1 (panel g), email-EU (panel  h), congress-bills (panel i), senate-committees (panel j), email-Enron (panel k), house-committees (panel l),  music-review (panel m), senate-bills (panel n), algebra-questions (panel o) and geometry-questions (panel p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  6  2  4  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='00  Cm(i)/km  max  a  R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='08  R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='7  R = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='1  R = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  2  4  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='00  b  R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='05  R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='6  R = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  R = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='00  c  R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='02  R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='8  R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='1  R = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='00  d  R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='03  R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='7  R = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='4  R = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='00  Cm(i)/km  max  e  R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='4  R = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='1  R = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='8  R = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='00  f  R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='3  R = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  R = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='6  R = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='6  5  10  15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='00  g  R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='2  R = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='1  R = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='6  R = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='9  10  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='00  h  R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='009  R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  R = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='8  R = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='2  10  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='00  Cm(i)/km  max  i  R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='003  R = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='6  R = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='4  R = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  10  20  30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='00  j  R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  R = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='9  R = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='3  R = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  10  20  30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='00  k  R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='1  R = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='3  R = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  R = 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  0  25  50  75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='00  l  R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='2  R = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='1  R = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='4  R = 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  0  25  50  75  m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='00  Cm(i)/km  max  m  R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='03  R = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='8  R = 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  R = 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  0  50  100  m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='00  n  R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='1  R = 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='4  R = 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='4  R = 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  0  50  100  m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='00  o  R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='02  R = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='1  R = 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='4  R = 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  0  100  200  m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='00  p  R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='01  R = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='2  R = 196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  R = 229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  Supplementary Figure 5: m-shell index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' All panels show the normalized m-shell index function Cm(i)/km  max as a  function of m for four nodes: one node is selected randomly among the nodes in the class with highest  hyper-coreness R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' one node is selected randomly among the nodes in the class with smallest hyper-coreness R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' the  two remaining node are selected from intermediate hyper-coreness classes, so that the positions in the  hyper-coreness ranking of the four nodes are equispaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The following data sets are considered: LH10 (panel a),  Thiers13 (panel b), InVS15 (panel c), SFHH (panel d), LyonSchool (panel e), Mid1 (panel f), Elem1 (panel g),  email-EU (panel h), congress-bills (panel i), senate-committees (panel j), email-Enron (panel k),  house-committees (panel l), music-review (panel m), senate-bills (panel n), algebra-questions (panel o) and  geometry-questions (panel p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  7  0  50  0  2  4  6  R(i)  a  0  5  R  0  20  P(R)  0  200  0  2  4  6  b  0  5  R  0  50  P(R)  0  100  200  0  2  4  6  8  c  0  5  R  0  50  P(R)  0  200  400  0  2  4  6  8  d  0  5  R  0  100  P(R)  0  100  200  0  2  4  6  8  R(i)  e  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  R  0  50  P(R)  0  250  500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  f  0  10  R  0  200  P(R)  0  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='g  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content=' The following data sets are  considered: LH10 (panel a), Thiers13 (panel b), InVS15 (panel c), SFHH (panel d), LyonSchool (panel e), Mid1  (panel f), Elem1 (panel g), email-EU (panel h), congress-bills (panel i), senate-committees (panel j), email-Enron  (panel k), house-committees (panel l), music-review (panel m), senate-bills (panel n), algebra-questions (panel o)  and geometry-questions (panel p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content=' s-coreness centralities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' All panels show scatterplots of the  hyper-coreness R(i) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' the s-coreness S(i) for all nodes: the text-box reports the Pearson correlation coefficient ρ  of R(i) and S(i) and the Kendall’s τ coefficient of the corresponding node rankings (in all cases the p-value for  both the coefficients is p ≪ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content=' The following data sets are considered: LH10 (panel a), Thiers13 (panel b),  InVS15 (panel c), SFHH (panel d), LyonSchool (panel e), Mid1 (panel f), Elem1 (panel g), email-EU (panel h),  congress-bills (panel i), senate-committees (panel j), email-Enron (panel k), house-committees (panel l),  music-review (panel m), senate-bills (panel n), algebra-questions (panel o) and geometry-questions (panel p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='0  5 × 10−4  LyonSchool  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='0  5 × 10−5  Elem1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content=' 9-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The tables summarize the parameters of the higher-order  non-linear SIS contagion process considered for each data set in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='8  /T n  S-rank  R-rank  Supplementary Figure 9: Higher-order non-linear contagion process - SIS model - I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' All panels give, as a  heat-map as a function of k and m, the average fraction ⟨τ/T⟩ of time being infected in the SIS steady state  averaged over the nodes of the (k, m)-hyper-core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The insets represent τ/T averaged over the first n nodes  according to the coreness rankings as a function of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' All results are obtained by averaging the results of 103  numerical simulations, with a single random seed of infection and with an observation window T = 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The  following data sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f),  Elem1 (g), email-EU (h), congress-bills (i), senate-committees (j), email-Enron (k), house-committees (l),  music-review (m), senate-bills (n), algebra-questions (o) and geometry-questions (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The (λ, ν) values considered  for each data set are reported in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='90  /T  a  m = 2  m = 3  m = 4  m = 5  0  20  40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='7 b  m = 2  m = 3  m = 4  m = 5  0  20  40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='85 c  m = 2  m = 3  m = 4  m = 5  0  10  20  30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='80 d  m = 2  m = 3  m = 4  m = 5  0  100  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='88  /T  e  m = 2  m = 3  m = 4  m = 5  0  200  400  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='82  f  m = 2  m = 4  m = 6  m = 8  0  200  400  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='35 g  m = 2  m = 4  m = 6  m = 8  0  50  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='60 h  m = 2  m = 6  m = 10  m = 14  0  500  1000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='80 k  m = 2  m = 4  m = 6  m = 8  5  10  15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='85  l  m = 2  m = 15  m = 41  m = 54  0  20  40  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='8  /T  m  m = 2  m = 10  m = 18  m = 32  0  500  1000  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='80  n  m = 2  m = 18  m = 34  m = 50  0  20  40  60  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='8  o  m = 2  m = 8  m = 14  m = 20  0  25  50  75  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='7  p  m = 2  m = 15  m = 28  m = 41  Supplementary Figure 10: Higher-order non-linear contagion process - SIS model - II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In all panels the  average fraction ⟨τ/T⟩ of time being infected in the SIS steady state averaged over the nodes of the  (k, m)-hyper-core is shown as a function of k at fixed values of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' All results are obtained by averaging the results  of 103 numerical simulations, with a single random seed of infection and with an observation window T = 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  The following data sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f),  Elem1 (g), email-EU (h), congress-bills (i), senate-committees (j), email-Enron (k), house-committees (l),  music-review (m), senate-bills (n), algebra-questions (o) and geometry-questions (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The (λ, ν) values considered  for each data set are reported in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  13  2  4  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='90  /T  a  k = 1  k = 10  k = 20  k = 30  2  4  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content=' In all panels the  average fraction ⟨τ/T⟩ of time being infected in the SIS steady state averaged over the nodes of the  (k, m)-hyper-core is shown as a function of m at fixed values of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='  The following data sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f),  Elem1 (g), email-EU (h), congress-bills (i), senate-committees (j), email-Enron (k), house-committees (l),  music-review (m), senate-bills (n), algebra-questions (o) and geometry-questions (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content=' All panels show, as  a function of k and m through a heat-map, the average epidemic final-size ⟨R∞⟩ produced by seeding the SIR  process in a single seed belonging to the (k, m)-hyper-core (averaged over all nodes of the hyper-core).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The insets  represent, as a function of n, R∞ averaged over the first n nodes according to coreness rankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' All results are  obtained by averaging the results of 300 numerical simulations for each seed (except for the congress-bills data set  which is the result of 10 simulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The following data sets are considered: LH10 (a), Thiers13 (b), InVS15 (c),  SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), email-EU (h), congress-bills (i), senate-committees (j),  email-Enron (k), house-committees (l), music-review (m), senate-bills (n), algebra-questions (o) and  geometry-questions (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The (λ, ν) values considered for each data set are reported in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='68  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='70  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='a  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='120  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='140  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='60  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='350  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='1200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='1300  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='1400  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='60  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='70  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='80 j  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 17  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 22  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='60  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='150  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='250 l  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 41  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 54  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='400  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='600  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='700  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='800  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='900  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 18  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='240  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='250  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='260  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='270  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='280  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='290 n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 18  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 34  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='60  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='250  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='300  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='350  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='o  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='75  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='300  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='350  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='400  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='450  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='550 p  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 28  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 41  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='Supplementary Figure 13: Higher-order non-linear contagion process - SIR model - II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In all panels the  average epidemic final-size ⟨R∞⟩ produced by seeding the SIR process in a single seed belonging to the  (k, m)-hyper-core (averaged over all nodes of the hyper-core) is shown as a function of k at fixed values of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' All  results are obtained by averaging the results of 300 numerical simulations for each seed (except for the  congress-bills data set which is the result of 10 simulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The following data sets are considered: LH10 (a),  Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), email-EU (h), congress-bills (i),  senate-committees (j), email-Enron (k), house-committees (l), music-review (m), senate-bills (n),  algebra-questions (o) and geometry-questions (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The (λ, ν) values considered for each data set are reported in  Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  16  2  4  6  58  60  62  64  66  68  70  R  a  k = 1  k = 10  k = 20  k = 30  2  4  6  80  90  100  110  120  130  b  k = 1  k = 10  k = 20  k = 30  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  40  60  80  100 c  k = 1  k = 10  k = 20  k = 30  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  330  340  350  360  370  380  390 d  k = 1  k = 10  k = 20  k = 30  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='260  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='250  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='300  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='350  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='o  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='k = 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='k = 20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='k = 30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='300  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='350  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='400  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='450  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='550  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='p  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='k = 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='k = 10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='k = 20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='k = 30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='Supplementary Figure 14: Higher-order non-linear contagion process - SIR model - III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In all panels the  average epidemic final-size ⟨R∞⟩ produced by seeding the SIR process in a single seed belonging to the  (k, m)-hyper-core (averaged over all nodes of the hyper-core) is shown as a function of m at fixed values of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' All  results are obtained by averaging the results of 300 numerical simulations for each seed (except for the  congress-bills data set which is the result of 10 simulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The following data sets are considered: LH10 (a),  Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), email-EU (h), congress-bills (i),  senate-committees (j), email-Enron (k), house-committees (l), music-review (m), senate-bills (n),  algebra-questions (o) and geometry-questions (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The (λ, ν) values considered for each data set are reported in  Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  17  Supplementary Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Threshold higher-order contagion process  Here we consider another spreading process in which multi-body interactions drive the infection through a thresh-  old effect and group contagion [2, 4]: the threshold higher-order contagion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' We consider both the SIR and  SIS epidemic models on static hypergraphs: for each hyperedge of size m in which i individuals are in the state I,  if the fraction of infected individuals i/m is larger or equal to a threshold θ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' if i ≥ ⌈θm⌉, a group infection is  activated at rate λ in which the susceptible nodes in the hyperedge become all infected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Note that if we consider a  single seed of infection: for θ ≤ 1/M the group infection is activated in all the hyperedges containing the seed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' for  θ = 1/m the spreading is activated only in the hyperedges containing the seed that have size larger or equal to m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  for θ > 1/2 the spreading is inhibited since more than one infected node is required to activate the infection in all  hyperedges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' I individuals recover independently at constant rate µ, becoming either S (SIS model) or R (SIR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  We perform numerical simulations of this process, for both SIS and SIR models, on empirical hypergraphs: the  simulation procedures are analogous to those described in the main text for the higher-order non-linear contagion  process (see Methods), since the two processes only differ in the infection mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In the threshold higher-order  contagion, for each time-step ∆t, given a hyperedge of size m containing i infected nodes, if i ≥ ⌈θm⌉ a group  infection process is activated with probability λ and all susceptible nodes in the hyperedge are infected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Thus, in  each time-step each of the interaction groups respecting the condition i ≥ ⌈θm⌉ produces a group infection process  with probability λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Therefore, also in this case we quantify the ”spreading power” of each node considered separately as seed for the  SIR model and the nodes on which the epidemic is mainly localized in the steady state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' the nodes that drive  and sustain the process, for the SIS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' 15-20 are shown the results of these simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  data set  θ  λ  LH10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='005  Thiers13  1/7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='001  InVS15  1/10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='001  SFHH  1/10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='001  LyonSchool  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='001  Mid1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='001  Elem1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='001  email-EU  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='001  data set  θ  λ  congress-bills  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='001  senate-committees  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='01  email-Enron  1/37  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='01  house-committees  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='01  music-review  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='01  senate-bills  1/99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0001  algebra-questions  1/107  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='001  geometry-questions  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='001  Supplementary Table IV: Parameters for Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' 15-17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The tables summarize the parameters of the threshold  higher-order SIS contagion process considered for each data set in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' 15-17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  data set  θ  λ  LH10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='01  Thiers13  1/7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='001  InVS15  1/10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='01  SFHH  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='01  LyonSchool  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='01  Mid1  1/13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='001  Elem1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='001  email-EU  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='001  data set  θ  λ  congress-bills  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='001  senate-committees  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='01  email-Enron  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='01  house-committees  1/82  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='01  senate-bills  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='4  /T n  S-rank  R-rank  Supplementary Figure 15: Threshold higher-order contagion process - SIS model - I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' All panels give, as a  heat-map as a function of k and m, the average fraction ⟨τ/T⟩ of time being infected in the SIS steady state  averaged over the nodes of the (k, m)-hyper-core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The insets represent τ/T averaged over the first n nodes  according to the coreness rankings as a function of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' All results are obtained by averaging the results of 103  numerical simulations, with a single random seed of infection and with an observation window T = 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The  following data sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f),  Elem1 (g), email-EU (h), congress-bills (i), senate-committees (j), email-Enron (k), house-committees (l),  music-review (m), senate-bills (n), algebra-questions (o) and geometry-questions (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The values (λ,θ) values  considered for each data set are summarized in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  19  0  20  40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='60 n  m = 2  m = 18  m = 34  m = 50  0  20  40  60  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='5 o  m = 2  m = 8  m = 14  m = 20  0  25  50  75  k  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='4  p  m = 2  m = 15  m = 28  m = 41  Supplementary Figure 16: Threshold higher-order contagion process - SIS model - II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In all panels the  average fraction ⟨τ/T⟩ of time being infected in the steady state averaged over the nodes of the (k, m)-hyper-core  is shown as a function of k at fixed values of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' All results are obtained by averaging the results of 103 numerical  simulations, with a single random seed of infection and with an observation window T = 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The following data  sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g),  email-EU (h), congress-bills (i), senate-committees (j), email-Enron (k), house-committees (l), music-review (m),  senate-bills (n), algebra-questions (o) and geometry-questions (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The values (λ,θ) values considered for each  data set are summarized in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  20  2  4  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='85  /T  a  k = 1  k = 10  k = 20  k = 30  2  4  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='06  b  k = 1  k = 10  k = 20  k = 30  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='35  p  k = 1  k = 10  k = 20  k = 30  Supplementary Figure 17: Threshold higher-order contagion process - SIS model - III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In all panels the  average fraction ⟨τ/T⟩ of time being infected in the steady state averaged over the nodes of the (k, m)-hyper-core  is shown as a function of m at fixed values of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' All results are obtained by averaging the results of 103 numerical  simulations, with a single random seed of infection and with an observation window T = 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The following data  sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g),  email-EU (h), congress-bills (i), senate-committees (j), email-Enron (k), house-committees (l), music-review (m),  senate-bills (n), algebra-questions (o) and geometry-questions (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The values (λ,θ) values considered for each  data set are summarized in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content=' All panels show, as  a function of k and m through a heat-map, the average epidemic final-size ⟨R∞⟩ produced by seeding the SIR  process in a single seed belonging to the (k, m)-hyper-core (averaged over all nodes of the hyper-core).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The insets  represent, as a function of n, R∞ averaged over the first n nodes according to coreness rankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' All results are  obtained by averaging the results of 300 numerical simulations for each seed (except for the congress-bills data set  which is the result of 10 simulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The following data sets are considered: LH10 (a), Thiers13 (b), InVS15 (c),  SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), email-EU (h), congress-bills (i), senate-committees (j),  email-Enron (k), house-committees (l), music-review (m), senate-bills (n), algebra-questions (o) and  geometry-questions (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='600  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='700  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='800  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='l  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 41  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 54  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='400  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='600  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='700  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='800  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 18  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='60  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='70  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='80 n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 18  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 34  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='60  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='60  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='70 o  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='75  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='60  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='120  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='140 p  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 28  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='m = 41  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='Supplementary Figure 19: Threshold higher-order contagion process - SIR model - II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In all panels the  average epidemic final-size ⟨R∞⟩ produced by seeding the SIR process in a single seed belonging to the  (k, m)-hyper-core (averaged over all nodes of the hyper-core) is shown as a function of k at fixed values of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' All  results are obtained by averaging the results of 300 numerical simulations for each seed (except for the  congress-bills data set which is the result of 10 simulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The following data sets are considered: LH10 (a),  Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), email-EU (h), congress-bills (i),  senate-committees (j), email-Enron (k), house-committees (l), music-review (m), senate-bills (n),  algebra-questions (o) and geometry-questions (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The values (λ,θ) values considered for each data set are  summarized in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  23  2  4  6  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  R  a  k = 1  k = 10  k = 20  k = 30  2  4  6  6  7  8  9  10  b  k = 1  k = 10  k = 20  k = 30  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  160  170  180  190  c  k = 1  k = 10  k = 20  k = 30  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  280  300  320  340  d  k = 1  k = 10  k = 20  k = 30  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='e  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='k = 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='k = 10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='k = 20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='k = 30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='470  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='480  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='490  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='510  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='f  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='k = 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='k = 30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='k = 20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='k = 30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='m  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='0  n  k = 1  k = 10  k = 20  k = 30  0  50  100  m  20  30  40  50  60  o  k = 1  k = 10  k = 20  k = 30  0  100  200  m  40  60  80  100  p  k = 1  k = 10  k = 20  k = 30  Supplementary Figure 20: Threshold higher-order contagion process - SIR model - III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In all panels the  average epidemic final-size ⟨R∞⟩ produced by seeding the SIR process in a single seed belonging to the  (k, m)-hyper-core (averaged over all nodes of the hyper-core) is shown as a function of m at fixed values of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' All  results are obtained by averaging the results of 300 numerical simulations for each seed (except for the  congress-bills data set which is the result of 10 simulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The following data sets are considered: LH10 (a),  Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), email-EU (h), congress-bills (i),  senate-committees (j), email-Enron (k), house-committees (l), music-review (m), senate-bills (n),  algebra-questions (o) and geometry-questions (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The values (λ,θ) values considered for each data set are  summarized in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  24  Supplementary Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Higher-order naming-game (NG) process  data set  β  p  tmax  T  InVS15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='41  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='8 × 10−2  105  104  Mid1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='59  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='5 × 10−2  105  104  email-EU  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='52  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='2 × 10−3  5 × 105  5 × 104  congress-bills  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='59  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='4 × 10−2  5 × 105  5 × 104  house-committees  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='55  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='0  nA(t)  x  Random  R-rank  s-rank  Supplementary Figure 21:  Higher-order NG process - Union rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In the first three panels of each row the  stationary fraction n∗  A of nodes supporting only the name A is shown as a function of the fraction of committed nodes p  and the agreement probability β through a heat-map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' We consider the union rule and the following data sets: InVS15 (first  row), Mid1 (second row), email-EU (third row), congress-bills (fourth row), house-committees (fifth row), music-reviews  (sixth row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' For each row, the committed nodes are selected through: the random seeding strategy in the first panel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' the  top s-coreness seeding strategy in the second panel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' the top hyper-coreness seeding strategy in the third panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In the  fourth panel we show the temporal dynamics of nA(t) for fixed β and p, whose values are reported in Table VI (cross  markers in the heatmaps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The minority take-over, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content=' n∗  A = 1, takes place over an area A of the explored parameter space:  in Table VII we report its value for the different strategies and data sets considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' All simulations are run until the  absorbing state with n∗  A = 1 is reached or the dynamics has evolved for tmax time steps and the stationary fraction n∗  A is  obtained by averaging over 100 values sampled in the last T time-steps (see Table VI for the tmax and T values for each  data set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
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+page_content='0  nA(t)  x  Random  R-rank  s-rank  Supplementary Figure 22:  Higher-order NG process - Unanimity rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In the first three panels of each row the  fraction n∗  A of nodes supporting only the name A is shown as a function of the fraction of committed nodes p and the  agreement probability β through a heat-map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' We consider the unanimity rule and the following data sets: InVS15 (first  row), Mid1 (second row), email-EU (third row), congress-bills (fourth row), house-committees (fifth row), music-reviews  (sixth row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' For each row, the committed nodes are selected through: the random seeding strategy in the first panel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' the  top s-coreness seeding strategy in the second panel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' the top hyper-coreness seeding strategy in the third panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' In the  fourth panel we show the temporal dynamics of nA(t) for fixed β and p, whose values are reported in Table VIII (cross  markers in the heatmaps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The minority take-over, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' n∗  A = 1, takes place over an area A of the explored parameter space:  in Table IX we report its value for the different strategies and data sets considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' All simulations are run until the  absorbing state with n∗  A = 1 is reached or the dynamics has evolved for tmax time steps and the stationary fraction n∗  A is  obtained by averaging over 100 values sampled in the last T time-steps (see Table VIII for the tmax and T values for each  data set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' The results refer to the median values obtained over 200 simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  28  [1] St-Onge, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Influential groups for seeding and sustaining nonlinear contagion in heterogeneous hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Communications Physics 5, 25 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  [2] de Arruda, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=', Petri, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=', Rodriguez, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' & Moreno, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Multistability, intermittency and hybrid transitions in social  contagion models on hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' arXiv preprint - arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='04273 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  [3] Iacopini, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=', Petri, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=', Baronchelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' & Barrat, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  Group interactions modulate critical mass dynamics in social  convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Communications Physics 5, 64 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content='  [4] de Arruda, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=', Petri, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' & Moreno, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Social contagion models on hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}
+page_content=' Research 2, 023032 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'}