DOCSLIB.ORG

A Scalable Community Detection Algorithm for Large Graphs Using Stochastic Block Models Chengbin Peng1, Zhihua Zhang2, Ka-Chun Wong3, Xiangliang Zhang1, David E

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Scalable Community Detection Algorithm for Large Graphs Using Stochastic Block Models Chengbin Peng1, Zhihua Zhang2, Ka-Chun Wong3, Xiangliang Zhang1, David E Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence (IJCAI 2015) A Scalable Community Detection Algorithm for Large Graphs Using Stochastic Block Models Chengbin Peng1, Zhihua Zhang2, Ka-Chun Wong3, Xiangliang Zhang1, David E. Keyes1 1King Abdullah University of Science and Technology, Thuwal, Saudi Arabia 2Shanghai Jiao Tong University, Shanghai, China 3City University of Hong Kong, Hong Kong, China fchengbin.peng, xiangliang.zhang, [email protected], [email protected], [email protected] Abstract for large graphs, for example, in handling community hetero- geneity [Nadler and Galun, 2006; Fortunato and Barthelemy, Community detection in graphs is widely used in 2007; Xiang and Hu, 2012]. Alternatively, model-based social and biological networks, and the stochastic methods can produce more reliable and accurate results when block model is a powerful probabilistic tool for de- the model assumptions are in accordance with the real graphs. scribing graphs with community structures. How- Stochastic block models (SBMs) are among the important ever, in the era of “big data,” traditional inference probabilistic tools describing the connectivity relationship be- algorithms for such a model are increasingly lim- tween pairs of nodes [Holland et al., 1983], and have re- ited due to their high time complexity and poor ceived considerable attention both in theoretical [Celisse et scalability. In this paper, we propose a multi-stage al., 2012] and application domains [Daudin et al., 2008]. maximum likelihood approach to recover the latent Many algorithms have been proposed to infer the param- parameters of the stochastic block model, in time eters of SBMs, such as Bayesian estimation [Hofman and linear with respect to the number of edges. We Wiggins, 2008] and nuclear norm minimization [Chen et al., also propose a parallel algorithm based on message 2012]. Bayesian estimation defines the prior distributions on passing. Our algorithm can overlap communica- latent variables (community labels of nodes) and maximizes tion and computation, providing speedup without the posterior distribution when a graph is given [Hofman and compromising accuracy as the number of proces- Wiggins, 2008]. Nuclear norm minimization of a matrix min- sors grows. For example, to process a real-world imizes the sum of its singular values, and their algorithm is graph with about 1.3 million nodes and 10 million verified on graphs containing one thousand nodes. edges, our algorithm requires about 6 seconds on In this work, we propose a multi-stage likelihood maxi- 64 cores of a contemporary commodity Linux clus- mization algorithm based on SBMs, which has three advan- ter. Experiments demonstrate that the algorithm tages: can produce high quality results on both bench- • Speed: We devise an algorithm based on coordinate de- mark and real-world graphs. An example of finding scent and use approximations to simplify computations. more meaningful communities is illustrated conse- Our algorithm runs in linear time with respect to the quently in comparison with a popular modularity number of edges. maximization algorithm. • Scalability: We propose a parallel algorithm based on a message-passing, which tends to be the most 1 Introduction portable and performance-consistent parallel program- Community structures, in which nodes belonging to the same ming model for a variety of memory structures. We community are more densely connected to each other than overlap the communication and computation in the al- externally, prevail in real graphs. Finding those structures gorithm, so that for large graphs, it can achieve signifi- can be beneficial in many fields, such as finding protein cant speedup proportional to the number of processors. complexes in biological networks and topical or disciplinary To the best of our knowledge, it is the first parallel algo- groups in collaborative networks [Fortunato and Castellano, rithm for inferring SBMs. 2012; Kemp et al., 2006; Ansotegui´ et al., 2012]. • Quality: The algorithm can produce high-quality re- In this work, we consider non-overlapping community sults. In the initialization, it considers each node as structures. Many community detection algorithms handle one community and then employs a multi-stage iterative such a problem [Newman, 2004; Raghavan et al., 2007; strategy to construct larger communities gradually. It Blondel et al., 2008; Fortunato and Castellano, 2012; Liu et outperforms traditional community detection algorithms al., 2013; Wickramaarachchi et al., 2014; Staudt and Mey- empirically as measured by normalized mutual infor- erhenke, 2015]. However, they come along with limitations mation (NMI) [Danon et al., 2005] with respect to the 2090 ground-truth. Experiments on real-world graphs show 3.1 Estimation Algorithm that our algorithm can produce more meaningful com- Given W , the maximum likelihood estimates of (B; Z) are munities with good quality. defined as n argmax L(B; ZjW ) 2 Stochastic Block Model B;Z X = log[(1−W ) + (2W −1)Θ ] In this work, we develop a community detection algorithm ij ij ij based on a stochastic block model (SBM). We define N as i6=j X T o the number of nodes and M as the number of undirected and = log[(1−Wij) + (2Wij−1)(ZBZ )ij] ; (2) unweighted edges connecting those nodes. Each node be- i6=j longs to one of K blocks, and we use Z 2 f0; 1gN×K to P represent the block labels. That is, Zir = 1 means node subject to 0 ≤ Bij ≤ 1, Zij 2 f0; 1g, j Zij = 1. Roughly i belongs to block r and each row of Z contains only one speaking, we solve the above optimization problem by alter- nonzero entry. We also define a matrix B 2 [0; 1]K×K natively updating B and Z and using a community shrinking where Brk(r 6= k) represents the probability of connections and expanding strategy to improve accuracy. between nodes drawn from block r and k, respectively. If We first describe the alternating updating procedure. When r = k, Brk represents the probability of connections inside Z is fixed and B is considered as unknown, without loss the block. of generality, let β = Brk. If other entries are fixed, Using the matrices B and Z, we define a probability matrix L(B; ZjW ) = s log(β) +s ^log(1 − β) + C, where C is T P T P Θ = ZBZ . Then the adjacency matrix W of a sample a constant, s = ij Wij(Z:rZ:k)ij and s^ = ij(1 − network can be drawn from the following model: T Wij)(Z:rZ:k)ij. Taking the derivative of L with respect to β, Θij if Wij = 1; @L s s^ Pr(Wij) = (1) = − (3) 1 − Θij if Wij = 0; @β β 1 − β and setting the derivative to be zero, we have for i; j 2 f1; 2; ··· ;Ng and i 6= j, indicating that Wij is a s sample from the Bernoulli distribution with success rate Θij. β = : (4) Typically, the adjacency matrix W is available from the data s +s ^ set. Our primary purpose is to estimate Z. As the inter-block connection probabilities are small, we use a representative scalar value to replace the off-diagonal en- tries of B, which can be computed by counting all the inter- 3 Methodology community edges. Thus, the total time complexity of updat- ing B is O(N) + O(K) + O(M) = O(M). For a specific model, the likelihood is a function of the model parameters, describing the probability of obtaining the ob- Theorem 1 For fixed Z, the objective function L(B; ZjW ) served data with these parameters. The maximum likelihood achieves its global maximum if entries of B are updated ac- estimator is the setting of parameters that maximizes the like- cording to Eq. (4). lihood function. Proof When Z is fixed, because each entry of B can opti- As defined in Eq. (1), if only W is given, the log-likelihood mize the objective function independently, after updating all function is the entries by Eq. (4), the resulting B is a stationary point of X the objective function and each entry satisfies the constraints. L(B; ZjW ) = log Pr(Wij) By taking the second derivative of the objective function, i6=j we have X = log[(1 − W ) + (2W − 1)Θ ]: 2 ij ij ij @ L s s^ i6=j 2 = − 2 − 2 < 0: (5) @β s β (1−β) s β= s+^s β= s+^s It is very time consuming to maximize such a likelihood func- Z β tion directly through traditional optimization methods (for ex- Therefore, when is given, the objective function at ample, branch-and-bound) for large graphs in which there are (determined by Eq. (4)) is a global maximum. As each entry B B at least NK unknown variables. of is irrelevant to each other, we can update by Eq. (4) sequentially. For the sake of speed and scalability, we propose a fast al- gorithm that updates B and Z in turn to maximize the objec- When B is fixed, we use the block coordinate descent tive function L(B; ZjW ), and use a multi-stage framework method to update Z row by row. When updating the first Z to help the solution be close to the global optimum. We also in ZBZT in Eq. (2), the algorithm keeps the second Z as its develop a parallel implementation to make the model more previous estimate Z(t−1). Then, the likelihood function can scalable. be locally maximized by setting all the elements in the row to 2091 0 but Zirmax = 1, where rmax is chosen by of communities to K. When K is unknown, the algorithm proceeds until no more merging is possible. X rmax = argmax log (1 − Wij) This approach reduces the “collision” probability in the ini- r j6=i tialization significantly, where a “collision” is the situation that two true communities both have most of the nodes in one + (2W − 1)(B[Z(t−1)]T ) : (6) ij rj temporary community.
Load more

Recommended publications
  • Arxiv:1705.10225V8 [Stat.ML] 6 Feb 2020
    Bayesian stochastic blockmodelinga Tiago P. Peixotoy Department of Mathematical Sciences and Centre for Networks and Collective Behaviour, University of Bath, United Kingdom and ISI Foundation, Turin, Italy This chapter provides a self-contained introduction to the use of Bayesian inference to ex- tract large-scale modular structures from network data, based on the stochastic blockmodel (SBM), as well as its degree-corrected and overlapping generalizations. We focus on non- parametric formulations that allow their inference in a manner that prevents overfitting, and enables model selection. We discuss aspects of the choice of priors, in particular how to avoid underfitting via increased Bayesian hierarchies, and we contrast the task of sampling network partitions from the posterior distribution with finding the single point estimate that maximizes it, while describing efficient algorithms to perform either one. We also show how inferring the SBM can be used to predict missing and spurious links, and shed light on the fundamental limitations of the detectability of modular structures in networks. arXiv:1705.10225v8 [stat.ML] 6 Feb 2020 aTo appear in “Advances in Network Clustering and Blockmodeling,” edited by P. Doreian, V. Batagelj, A. Ferligoj, (Wiley, New York, 2019 [forthcoming]). y [email protected] 2 CONTENTS I. Introduction 3 II. Structure versus randomness in networks 3 III. The stochastic blockmodel (SBM) 5 IV. Bayesian inference: the posterior probability of partitions 7 V. Microcanonical models and the minimum description length principle (MDL) 11 VI. The “resolution limit” underfitting problem, and the nested SBM 13 VII. Model variations 17 A. Model selection 18 B. Degree correction 18 C.
    [Show full text]
  • On Community Structure in Complex Networks: Challenges and Opportunities
    On community structure in complex networks: challenges and opportunities Hocine Cherifi · Gergely Palla · Boleslaw K. Szymanski · Xiaoyan Lu Received: November 5, 2019 Abstract Community structure is one of the most relevant features encoun- tered in numerous real-world applications of networked systems. Despite the tremendous effort of a large interdisciplinary community of scientists working on this subject over the past few decades to characterize, model, and analyze communities, more investigations are needed in order to better understand the impact of community structure and its dynamics on networked systems. Here, we first focus on generative models of communities in complex networks and their role in developing strong foundation for community detection algorithms. We discuss modularity and the use of modularity maximization as the basis for community detection. Then, we follow with an overview of the Stochastic Block Model and its different variants as well as inference of community structures from such models. Next, we focus on time evolving networks, where existing nodes and links can disappear, and in parallel new nodes and links may be introduced. The extraction of communities under such circumstances poses an interesting and non-trivial problem that has gained considerable interest over Hocine Cherifi LIB EA 7534 University of Burgundy, Esplanade Erasme, Dijon, France E-mail: hocine.cherifi@u-bourgogne.fr Gergely Palla MTA-ELTE Statistical and Biological Physics Research Group P´azm´any P. stny. 1/A, Budapest, H-1117, Hungary E-mail: [email protected] Boleslaw K. Szymanski Department of Computer Science & Network Science and Technology Center Rensselaer Polytechnic Institute 110 8th Street, Troy, NY 12180, USA E-mail: [email protected] Xiaoyan Lu Department of Computer Science & Network Science and Technology Center Rensselaer Polytechnic Institute 110 8th Street, Troy, NY 12180, USA E-mail: [email protected] arXiv:1908.04901v3 [physics.soc-ph] 6 Nov 2019 2 Cherifi, Palla, Szymanski, Lu the last decade.
    [Show full text]
  • Community Detection with Dependent Connectivity∗
    Submitted to the Annals of Statistics arXiv: arXiv:0000.0000 COMMUNITY DETECTION WITH DEPENDENT CONNECTIVITY∗ BY YUBAI YUANy AND ANNIE QUy University of Illinois at Urbana-Champaigny In network analysis, within-community members are more likely to be connected than between-community members, which is reflected in that the edges within a community are intercorrelated. However, existing probabilistic models for community detection such as the stochastic block model (SBM) are not designed to capture the dependence among edges. In this paper, we propose a new community detection approach to incorporate intra-community dependence of connectivities through the Bahadur representation. The pro- posed method does not require specifying the likelihood function, which could be intractable for correlated binary connectivities. In addition, the pro- posed method allows for heterogeneity among edges between different com- munities. In theory, we show that incorporating correlation information can achieve a faster convergence rate compared to the independent SBM, and the proposed algorithm has a lower estimation bias and accelerated convergence compared to the variational EM. Our simulation studies show that the pro- posed algorithm outperforms the popular variational EM algorithm assuming conditional independence among edges. We also demonstrate the application of the proposed method to agricultural product trading networks from differ- ent countries. 1. Introduction. Network data has arisen as one of the most common forms of information collection. This is due to the fact that the scope of study not only focuses on subjects alone, but also on the relationships among subjects. Networks consist of two components: (1) nodes or ver- tices corresponding to basic units of a system, and (2) edges representing connections between nodes.
    [Show full text]
  • Developing a Credit Scoring Model Using Social Network Analysis
    Developing a Credit Scoring Model Using Social Network Analysis Ahmad Abd Rabuh UP821483 This thesis is submitted in fulfilment of the requirements for the award of the degree of Doctor of Philosophy in the School of Business and Law at the University of Portsmouth November 2020 1 Author’s Declaration Whilst registered as a candidate for the above degree, I have not been registered for any other research award. The results and conclusions embodied in this thesis are the work of the named candidate and have not been submitted for any other academic award. Signature: Name: Ahmad Abd Rabuh Date: the 30th of April 2020 Word Count: 63,809 2 Acknowledgements As I witness this global pandemic, my greatest gratitude goes to my mother, Hanan, for her moral support and encouragement through the good, the bad and the ugly. During the period of my PhD revision, I received mentoring and help from my future lifetime partner, Kawthar. Also, I have been blessed with the help of wonderful people from Scholars at Risk, Rose Anderson and Sarina Rosenthal, who assisted me in the PhD application and scholarship processes. Additionally, I cannot be thankful enough to the Associate Dean of Research, Andy Thorpe, who supported me in every possible way during my time at the University of Portsmouth. Finally, many thanks to my supervisors, Mark Xu and Renatas Kizys for their continuous guidance. 3 Developing a Credit Scoring Model Using Social Network Analysis Table of Contents Abstract ......................................................................................................................................... 10 CHAPTER 1: INTRODUCTION ................................................................................................. 12 1.1. Research Questions ........................................................................................................ 16 1.2. Aims and Objectives .....................................................................................................
    [Show full text]
  • Exploratory Analysis of Networks
    Exploratory Analysis of Networks James D. Wilson Data Institute Conference 2017 James D. Wilson (USF) Exploratory Analysis of Networks 1 / 35 Summarizing Networks Goals 1 Provide one number summaries of network 2 Develop hypotheses about observed data 3 Motivate predictive models 4 Augment standard multivariate analyses James D. Wilson (USF) Exploratory Analysis of Networks 2 / 35 At first glance: Network Summary Statistics Measures of Connectivity Degree: number of edges incident to each node (popularity) For directed networks, we consider In- and Out-Degree Geodesic distance: shortest path between two nodes Diameter: longest geodesic distance Clustering coefficient: fraction of triangles to total triples Reciprocity: fraction of reciprocated ties (directed graphs only) James D. Wilson (USF) Exploratory Analysis of Networks 3 / 35 Examples in Social Networks James D. Wilson (USF) Exploratory Analysis of Networks 4 / 35 At first glance: Network Summary Statistics Measures of Nodal Influence / Centrality Centrality: how ”central” is a node in the observed network? Degree: popularity of a node Eigenvector: self- neighbor- importance Betweenness: extent+ to which a node lies on the shortest path between two nodes Closeness: mean distance from node to all other vertices Authorities: vertices with useful information on a topic Hubs: vertices that point to authorities James D. Wilson (USF) Exploratory Analysis of Networks 5 / 35 Examples Internet Search PageRank Algorithm Popular web-search scoring and search algorithm (Google) Score based on eigenvector centrality HITS Algorithm (Kleinberg, 1999) Hyperlink-induced topic search Uses hub and authority scores as basis for measuring importance of webpages James D. Wilson (USF) Exploratory Analysis of Networks 6 / 35 Example: Recommendation Systems Figure: eye2data.blogspot.com Want individual (node) with most influence James D.
    [Show full text]
  • On Spectral Embedding Performance and Elucidating Network Structure in Stochastic Block Model Graphs
    On spectral embedding performance and elucidating network structure in stochastic block model graphs Joshua Cape and Minh Tang and Carey E. Priebe Department of Applied Mathematics and Statistics Johns Hopkins University August 16, 2018 Abstract Statistical inference on graphs often proceeds via spectral methods involving low- dimensional embeddings of matrix-valued graph representations, such as the graph Laplacian or adjacency matrix. In this paper, we analyze the asymptotic information- theoretic relative performance of Laplacian spectral embedding and adjacency spectral embedding for block assignment recovery in stochastic block model graphs by way of Chernoff information. We investigate the relationship between spectral embedding per- formance and underlying network structure (e.g. homogeneity, affinity, core-periphery, (un)balancedness) via a comprehensive treatment of the two-block stochastic block model and the class of K-block models exhibiting homogeneous balanced affinity struc- ture. Our findings support the claim that, for a particular notion of sparsity, loosely speaking, “Laplacian spectral embedding favors relatively sparse graphs, whereas adja- cency spectral embedding favors not-too-sparse graphs.” We also provide evidence in support of the claim that “adjacency spectral embedding favors core-periphery network structure.” 2010 Mathematics Subject Classification: Primary 62H30; Secondary 62B10. arXiv:1808.04855v1 [math.ST] 14 Aug 2018 Keywords: Statistical network analysis; random graphs; stochastic block model; Laplacian spectral embed- ding; adjacency spectral embedding; Chernoff information; vertex clustering. 1 1 Preface The stochastic block model (SBM) (Holland et al., 1983) is a simple yet ubiquitous network model capable of capturing community structure that has been widely studied via spectral methods in the mathematics, statistics, physics, and engineering communities.
    [Show full text]
  • Link Prediction in Large-Scale Complex Networks (Application to Bibliographical Networks)
    N◦ attribué par la bibliothèque | _ | _ | _ | _ | _ | _ | _ | _ | _ | _ | Université Paris Nord Doctoral Thesis Link Prediction in Large-scale Complex Networks (Application to Bibliographical Networks) Author: Manisha Pujari A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science in the research team Apprentissage Artificiel et Applications LIPN CNRS UMR-7030 Jury: Reviewer: Céline Robardet Professor INSA Lyon Bénédicte le Grand Professor Université Paris 1 Panthéon Sorbonne Examiner: Aldo Gangemi Professor SPC, Université Paris 13 Christophe Prieur Associate Professor, HDR Université Paris Diderot Director: Céline Rouveirol Professor SPC, Université Paris 13 Supervisor: Rushed Kanawati Associate Professor SPC, Université Paris 13 “ The more I learn, the more I realize how much I don’t know. ” Albert Einstein Abstract Link Prediction in Large-scale Complex Networks (Application to Bibliographical Networks) In this work, we are interested to tackle the problem of link prediction in complex net- works. In particular, we explore topological dyadic approaches for link prediction. Dif- ferent topological proximity measures have been studied in the scientific literature for finding the probability of appearance of new links in a complex network. Supervised learning methods have also been used to combine the predictions made or information provided by different topological measures. They create predictive models using various topological measures. The problem of supervised learning for link prediction is a difficult problem especially due to the presence of heavy class imbalance. In this thesis, we search different alternative approaches to improve the performance of different dyadic approaches for link prediction.
    [Show full text]
  • Dynamic Stochastic Block Models, Clustering and Segmentations in Dynamic Graphs Marco Corneli
    Dynamic stochastic block models, clustering and segmentations in dynamic graphs Marco Corneli To cite this version: Marco Corneli. Dynamic stochastic block models, clustering and segmentations in dynamic graphs. Mathematics [math]. Université Paris 1 - Panthéon Sorbonne, 2017. English. tel-01644866 HAL Id: tel-01644866 https://hal.archives-ouvertes.fr/tel-01644866 Submitted on 22 Nov 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Université Paris 1 Pahntéon-Sorbonne Laboratoire SAMM Thèse présentée en première version en vu d’obtenir le grade de Docteur, spécialité "Mathématiques Appliquées" par Marco Corneli Dynamic stochastic block models, clustering and segmentation in dynamic graphs Thèse soutenue le 17 Novembre 2017 devant le jury composé de : M. Christophe Biernacki Université Lille 1 Sciences et Technologies (Rapporteur) M. Thomas Brendan Murphy University College Dublin (Rapporteur) M. Allou-Badara Samé IFSTTAR (Examinateur) Mme. Clémence Magnien CNRS (Examinatrice) Mme. Sophie Lèbre Université Paul Valéry Montpellier 3 (Examinatrice) M. Fabrice Rossi Université Paris 1 Panthéon-Sorbonne (Directeur) M. Pierre Latouche Université Paris 1 Panthéon-Sorbonne (Co-encadrant) Alla mia famiglia... Remerciements e remercie Fabrice Rossi et Pierre Latouche pour m’avoir fait confiance J et avoir encadré ce travail de thèse.
    [Show full text]
  • Efficient Inference in Stochastic Block Models with Vertex Labels
    1 Efficient inference in stochastic block models with vertex labels Clara Stegehuis and Laurent Massoulie´ Abstract—We study the stochastic block model with two connectivity matrix and d the average degree in the network. communities where vertices contain side information in the form Underneath the Kesten-Stigum threshold, no algorithm is able of a vertex label. These vertex labels may have arbitrary label to infer the community memberships better than a random distributions, depending on the community memberships. We analyze a linearized version of the popular belief propagation guess, even though a community structure may be present [14]. algorithm. We show that this algorithm achieves the highest In this setting, it is even impossible to discriminate between accuracy possible whenever a certain function of the network a graph generated by the stochastic block model and an parameters has a unique fixed point. Whenever this function has Erdos˝ Renyi´ random graph with the same average degree, multiple fixed points, the belief propagation algorithm may not even though a community structure is present [13]. Above the perform optimally. We show that increasing the information in the vertex labels may reduce the number of fixed points and Kesten-Stigum threshold, the communities can be efficiently hence lead to optimality of belief propagation. reconstructed [11], [15]. A popular algorithm for community detection is belief propagation [5] (BP). This algorithm starts with initial beliefs I. INTRODUCTION on the community memberships, and iterates until these beliefs converge to a fixed point. Above the Kesten-Stigum threshold, Many real-world networks contain community structures: a fixed point that is correlated with the true community mem- groups of densely connected nodes.
    [Show full text]
  • A Review of Stochastic Block Models and Extensions for Graph Clustering
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Newcastle University E-Prints Lee and Wilkinson Applied Network Science (2019) 4:122 Applied Network Science https://doi.org/10.1007/s41109-019-0232-2 REVIEW Open Access A review of stochastic block models and extensions for graph clustering Clement Lee1* and Darren J. Wilkinson2,3 *Correspondence: [email protected] Abstract 1Department of Mathematics and There have been rapid developments in model-based clustering of graphs, also known Statistics, Lancaster University, Lancaster, United Kingdom as block modelling, over the last ten years or so. We review different approaches and Full list of author information is extensions proposed for different aspects in this area, such as the type of the graph, the available at the end of the article clustering approach, the inference approach, and whether the number of groups is selected or estimated. We also review models that combine block modelling with topic modelling and/or longitudinal modelling, regarding how these models deal with multiple types of data. How different approaches cope with various issues will be summarised and compared, to facilitate the demand of practitioners for a concise overview of the current status of these areas of literature. Keywords: Model-based clustering, Stochastic block models, Mixed membership models, Topic modelling, Longitudinal modelling, Statistical inference Introduction Stochastic block models (SBMs) are an increasingly popular class of models in statistical analysis of graphs or networks. They can be used to discover or understand the (latent) structure of a network, as well as for clustering purposes.
    [Show full text]
  • Interpretable Stochastic Block Influence Model: Measuring Social Influence Among Homophilous Communities
    Interpretable Stochastic Block Influence Model: measuring social influence among homophilous communities YAN LENG, MIT Media Lab, McCombs School of Business at UT-Austin TARA SOWRIRAJAN, Harvard University, MIT Media Lab ALEX PENTLAND, MIT Media Lab Decision-making on networks can be explained by both homophily and social influence. While homophily drives the formation of communities with similar characteristics, social influence occurs both within and between communities. Social influence can be reasoned through role theory, which indicates that the influences among individuals depend on their roles and the behavior of interest. To operationalize these social science theories, we empirically identify the homophilous communities and use the community structures to capture the “roles”, which affect the particular decision-making processes. We propose a generative model named Stochastic Block Influence Model and jointly analyze both the network formation and the behavioral influence within and between different empirically-identified communities. To evaluate the performance and demonstrate the interpretability of our method, we study the adoption decisions of microfinance in an Indian village. We show that although individuals tend to form links within communities, there are strong positive and negative social influences between communities, supporting the weak tie theory. Moreover, we find that communities with shared characteristics are associated with positive influence. In contrast, the communities with a lack of overlap are associated with negative influence. Our framework facilitates the quantification of the influences underlying decision communities and is thus a useful tool for driving information diffusion, viral marketing, and technology adoptions. Key Words: Social influence; Homophily; Stochastic Block Model; Community structure; Generative model 1 INTRODUCTION We are living in an increasingly connected society [4, 26, 28, 37].
    [Show full text]
  • Barabasi-Albert Scale-Free Networks • Repeat Measurements Multiple Times and Plot Histograms of Assortativity
    Random Networks © Mario Freese Hiroki Sayama [email protected] Small-World Phenomenon “Bacon Number” • BN = 1 for those who co-starred with Kevin Bacon in a film • BN = 2 for those who co-starred with actors/actresses with BN=1 … • Mostly BN <= 3 !! • The largest finite BN = 8 !! 3 “Erdös Number” • EN = 1 for those who co-authored a paper with a Hungarian mathematician Paul Erdös (1913-1996) • EN = 2 for those who co-authored a paper with authors with EN=1 … • Mostly EN <= 7 !! • The largest finite EN = 13 !! FYI - Hiroki’s EN=4 (by Bing/Microsoft Academic Search) 4 3.5 degrees of separation in FB • https://research.facebook.com/blog/three-and-a- half-degrees-of-separation/ 5 “Small-world” phenomenon • Most real-world networks are remarkably “small” – Despite a huge number of nodes involved – Even if connections are relatively sparse • Why? 6 Random Networks Classical explanation: Erdös-Rényi random network model • A network made of N nodes • Each node pair is connected randomly and independently with probability p • A small characteristic path length is realized because of randomness – Number of nodes reachable from a single node within k steps increases exponentially with k 8 Exercise • Create and plot a few ER random networks using NetworkX • Measure their properties – Network density – Characteristic path length – Clustering coefficient – Degree distribution – etc. 9 Limitation of ER networks • ER random networks have very few loops or local clusters if connection probability is small • Real-world networks are often clustered with
    [Show full text]