On Community Structure in Complex Networks: Challenges and Opportunities
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Detecting Statistically Significant Communities
1 Detecting Statistically Significant Communities Zengyou He, Hao Liang, Zheng Chen, Can Zhao, Yan Liu Abstract—Community detection is a key data analysis problem across different fields. During the past decades, numerous algorithms have been proposed to address this issue. However, most work on community detection does not address the issue of statistical significance. Although some research efforts have been made towards mining statistically significant communities, deriving an analytical solution of p-value for one community under the configuration model is still a challenging mission that remains unsolved. The configuration model is a widely used random graph model in community detection, in which the degree of each node is preserved in the generated random networks. To partially fulfill this void, we present a tight upper bound on the p-value of a single community under the configuration model, which can be used for quantifying the statistical significance of each community analytically. Meanwhile, we present a local search method to detect statistically significant communities in an iterative manner. Experimental results demonstrate that our method is comparable with the competing methods on detecting statistically significant communities. Index Terms—Community Detection, Random Graphs, Configuration Model, Statistical Significance. F 1 INTRODUCTION ETWORKS are widely used for modeling the structure function that is able to always achieve the best performance N of complex systems in many fields, such as biology, in all possible scenarios. engineering, and social science. Within the networks, some However, most of these objective functions (metrics) do vertices with similar properties will form communities that not address the issue of statistical significance of commu- have more internal connections and less external links. -
Networks in Nature: Dynamics, Evolution, and Modularity
Networks in Nature: Dynamics, Evolution, and Modularity Sumeet Agarwal Merton College University of Oxford A thesis submitted for the degree of Doctor of Philosophy Hilary 2012 2 To my entire social network; for no man is an island Acknowledgements Primary thanks go to my supervisors, Nick Jones, Charlotte Deane, and Mason Porter, whose ideas and guidance have of course played a major role in shaping this thesis. I would also like to acknowledge the very useful sug- gestions of my examiners, Mark Fricker and Jukka-Pekka Onnela, which have helped improve this work. I am very grateful to all the members of the three Oxford groups I have had the fortune to be associated with: Sys- tems and Signals, Protein Informatics, and the Systems Biology Doctoral Training Centre. Their companionship has served greatly to educate and motivate me during the course of my time in Oxford. In particular, Anna Lewis and Ben Fulcher, both working on closely related D.Phil. projects, have been invaluable throughout, and have assisted and inspired my work in many different ways. Gabriel Villar and Samuel Johnson have been col- laborators and co-authors who have helped me to develop some of the ideas and methods used here. There are several other people who have gener- ously provided data, code, or information that has been directly useful for my work: Waqar Ali, Binh-Minh Bui-Xuan, Pao-Yang Chen, Dan Fenn, Katherine Huang, Patrick Kemmeren, Max Little, Aur´elienMazurie, Aziz Mithani, Peter Mucha, George Nicholson, Eli Owens, Stephen Reid, Nico- las Simonis, Dave Smith, Ian Taylor, Amanda Traud, and Jeffrey Wrana. -
Emerging Topics in Internet Technology: a Complex Networks Approach
1 Emerging Topics in Internet Technology: A Complex Networks Approach Bisma S. Khan 1, Muaz A. Niazi *,2 COMSATS Institute of Information Technology, Islamabad, Pakistan Abstract —Communication networks, in general, and internet technology, in particular, is a fast-evolving area of research. While it is important to keep track of emerging trends in this domain, it is such a fast-growing area that it can be very difficult to keep track of literature. The problem is compounded by the fast-growing number of citation databases. While other databases are gradually indexing a large set of reliable content, currently the Web of Science represents one of the most highly valued databases. Research indexed in this database is known to highlight key advancements in any domain. In this paper, we present a Complex Network-based analytical approach to analyze recent data from the Web of Science in communication networks. Taking bibliographic records from the recent period of 2014 to 2017 , we model and analyze complex scientometric networks. Using bibliometric coupling applied over complex citation data we present answers to co-citation patterns of documents, co-occurrence patterns of terms, as well as the most influential articles, among others, We also present key pivot points and intellectual turning points. Complex network analysis of the data demonstrates a considerably high level of interest in two key clusters labeled descriptively as “social networks” and “computer networks”. In addition, key themes in highly cited literature were clearly identified as “communication networks,” “social networks,” and “complex networks”. Index Terms —CiteSpace, Communication Networks, Network Science, Complex Networks, Visualization, Data science. -
Modularity in Static and Dynamic Networks
Modularity in static and dynamic networks Sarah F. Muldoon University at Buffalo, SUNY OHBM – Brain Graphs Workshop June 25, 2017 Outline 1. Introduction: What is modularity? 2. Determining community structure (static networks) 3. Comparing community structure 4. Multilayer networks: Constructing multitask and temporal multilayer dynamic networks 5. Dynamic community structure 6. Useful references OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Introduction: What is Modularity? OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon What is Modularity? Modularity (Community Structure) • A module (community) is a subset of vertices in a graph that have more connections to each other than to the rest of the network • Example social networks: groups of friends Modularity in the brain: • Structural networks: communities are groups of brain areas that are more highly connected to each other than the rest of the brain • Functional networks: communities are groups of brain areas with synchronous activity that is not synchronous with other brain activity OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Findings: The Brain is Modular Structural networks: cortical thickness correlations sensorimotor/spatial strategic/executive mnemonic/emotion olfactocentric auditory/language visual processing Chen et al. (2008) Cereb Cortex OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Findings: The Brain is Modular • Functional networks: resting state fMRI He et al. (2009) PLOS One OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Findings: The -
A Python Library to Model and Analyze Diffusion Processes Over Complex Networks
International Journal of Data Science and Analytics (2018) 5:61–79 https://doi.org/10.1007/s41060-017-0086-6 APPLICATIONS NDlib: a python library to model and analyze diffusion processes over complex networks Giulio Rossetti2 · Letizia Milli1,2 · Salvatore Rinzivillo2 · Alina Sîrbu1 · Dino Pedreschi1 · Fosca Giannotti2 Received: 19 October 2017 / Accepted: 11 December 2017 / Published online: 20 December 2017 © Springer International Publishing AG, part of Springer Nature 2017 Abstract Nowadays the analysis of dynamics of and on networks represents a hot topic in the social network analysis playground. To support students, teachers, developers and researchers, in this work we introduce a novel framework, namely NDlib,an environment designed to describe diffusion simulations. NDlib is designed to be a multi-level ecosystem that can be fruitfully used by different user segments. For this reason, upon NDlib, we designed a simulation server that allows remote execution of experiments as well as an online visualization tool that abstracts its programmatic interface and makes available the simulation platform to non-technicians. Keywords Social network analysis software · Epidemics · Opinion dynamics 1 Introduction be modeled as networks and, as such, analyzed. Undoubt- edly, such pervasiveness has produced an amplification in the In the last decades, social network analysis, henceforth SNA, visibility of network analysis studies thus making this com- has received increasing attention from several heterogeneous plex and interesting field one of the most widespread among fields of research. Such popularity was certainly due to the higher education centers, universities and academies. Given flexibility offered by graph theory: a powerful tool that the exponential diffusion reached by SNA, several tools were allows reducing countless phenomena to a common ana- developed to make it approachable to the wider audience pos- lytical framework whose basic bricks are nodes and their sible. -
Evolving Networks and Social Network Analysis Methods And
DOI: 10.5772/intechopen.79041 ProvisionalChapter chapter 7 Evolving Networks andand SocialSocial NetworkNetwork AnalysisAnalysis Methods and Techniques Mário Cordeiro, Rui P. Sarmento,Sarmento, PavelPavel BrazdilBrazdil andand João Gama Additional information isis available atat thethe endend ofof thethe chapterchapter http://dx.doi.org/10.5772/intechopen.79041 Abstract Evolving networks by definition are networks that change as a function of time. They are a natural extension of network science since almost all real-world networks evolve over time, either by adding or by removing nodes or links over time: elementary actor-level network measures like network centrality change as a function of time, popularity and influence of individuals grow or fade depending on processes, and events occur in net- works during time intervals. Other problems such as network-level statistics computation, link prediction, community detection, and visualization gain additional research impor- tance when applied to dynamic online social networks (OSNs). Due to their temporal dimension, rapid growth of users, velocity of changes in networks, and amount of data that these OSNs generate, effective and efficient methods and techniques for small static networks are now required to scale and deal with the temporal dimension in case of streaming settings. This chapter reviews the state of the art in selected aspects of evolving social networks presenting open research challenges related to OSNs. The challenges suggest that significant further research is required in evolving social networks, i.e., existent methods, techniques, and algorithms must be rethought and designed toward incremental and dynamic versions that allow the efficient analysis of evolving networks. Keywords: evolving networks, social network analysis 1. -
Homophily and Long-Run Integration in Social Networks∗
Homophily and Long-Run Integration in Social Networks∗ Yann Bramoulléy Sergio Currariniz Matthew O. Jacksonx Paolo Pin{ Brian W. Rogersk January 19, 2012 Abstract We study network formation in which nodes enter sequentially and form connections through a combination of random meetings and network-based search, as in Jackson and Rogers (2007). We focus on the impact of agents' heterogeneity on link patterns when connections are formed under type-dependent biases. In particular, we are concerned with how the local neighborhood of a node evolves as the node ages. We provide a surprising general result on long-run in- tegration whereby the composition of types in a node's neighborhood approaches the global type distribution, provided that the search part of the meeting process is unbiased. Integration, however, occurs only for suciently old nodes, while the aggregate distribution of connections still reects the bias of the random process. For a special case of the model, we analyze the form of these biases with regard to type-based degree distributions and group-level homophily patterns. Finally, we illustrate aspects of the model with an empirical application to data on citations in physics journals. JEL Codes: A14, D85, I21. ∗Following the suggestion of JET editors, this paper draws from two working papers developed independently: Bramoullé and Rogers (2010) and Currarini, Jackson and Pin (2010b). We gratefully acknowledge nancial support from the NSF under grant SES-0961481 and we thank Vincent Boucher for his research assistance. We also thank Habiba Djebbari, Andrea Galeotti, Sanjeev Goyal, James Moody, Betsy Sinclair, Bruno Strulovici, and Adrien Vigier, as well as numerous seminar participants. -
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. -
Multidimensional Network Analysis
Universita` degli Studi di Pisa Dipartimento di Informatica Dottorato di Ricerca in Informatica Ph.D. Thesis Multidimensional Network Analysis Michele Coscia Supervisor Supervisor Fosca Giannotti Dino Pedreschi May 9, 2012 Abstract This thesis is focused on the study of multidimensional networks. A multidimensional network is a network in which among the nodes there may be multiple different qualitative and quantitative relations. Traditionally, complex network analysis has focused on networks with only one kind of relation. Even with this constraint, monodimensional networks posed many analytic challenges, being representations of ubiquitous complex systems in nature. However, it is a matter of common experience that the constraint of considering only one single relation at a time limits the set of real world phenomena that can be represented with complex networks. When multiple different relations act at the same time, traditional complex network analysis cannot provide suitable an- alytic tools. To provide the suitable tools for this scenario is exactly the aim of this thesis: the creation and study of a Multidimensional Network Analysis, to extend the toolbox of complex network analysis and grasp the complexity of real world phenomena. The urgency and need for a multidimensional network analysis is here presented, along with an empirical proof of the ubiquity of this multifaceted reality in different complex networks, and some related works that in the last two years were proposed in this novel setting, yet to be systematically defined. Then, we tackle the foundations of the multidimensional setting at different levels, both by looking at the basic exten- sions of the known model and by developing novel algorithms and frameworks for well-understood and useful problems, such as community discovery (our main case study), temporal analysis, link prediction and more. -
Centrality in Modular Networks
Ghalmane et al. EPJ Data Science (2019)8:15 https://doi.org/10.1140/epjds/s13688-019-0195-7 REGULAR ARTICLE OpenAccess Centrality in modular networks Zakariya Ghalmane1,3, Mohammed El Hassouni1, Chantal Cherifi2 and Hocine Cherifi3* *Correspondence: hocine.cherifi@u-bourgogne.fr Abstract 3LE2I, UMR6306 CNRS, University of Burgundy, Dijon, France Identifying influential nodes in a network is a fundamental issue due to its wide Full list of author information is applications, such as accelerating information diffusion or halting virus spreading. available at the end of the article Many measures based on the network topology have emerged over the years to identify influential nodes such as Betweenness, Closeness, and Eigenvalue centrality. However, although most real-world networks are made of groups of tightly connected nodes which are sparsely connected with the rest of the network in a so-called modular structure, few measures exploit this property. Recent works have shown that it has a significant effect on the dynamics of networks. In a modular network, a node has two types of influence: a local influence (on the nodes of its community) through its intra-community links and a global influence (on the nodes in other communities) through its inter-community links. Depending on the strength of the community structure, these two components are more or less influential. Based on this idea, we propose to extend all the standard centrality measures defined for networks with no community structure to modular networks. The so-called “Modular centrality” is a two-dimensional vector. Its first component quantifies the local influence of a node in its community while the second component quantifies its global influence on the other communities of the network. -
A Network Approach to Define Modularity of Components In
A Network Approach to Define Modularity of Components Manuel E. Sosa1 Technology and Operations Management Area, in Complex Products INSEAD, 77305 Fontainebleau, France Modularity has been defined at the product and system levels. However, little effort has e-mail: [email protected] gone into defining and quantifying modularity at the component level. We consider com- plex products as a network of components that share technical interfaces (or connections) Steven D. Eppinger in order to function as a whole and define component modularity based on the lack of Sloan School of Management, connectivity among them. Building upon previous work in graph theory and social net- Massachusetts Institute of Technology, work analysis, we define three measures of component modularity based on the notion of Cambridge, Massachusetts 02139 centrality. Our measures consider how components share direct interfaces with adjacent components, how design interfaces may propagate to nonadjacent components in the Craig M. Rowles product, and how components may act as bridges among other components through their Pratt & Whitney Aircraft, interfaces. We calculate and interpret all three measures of component modularity by East Hartford, Connecticut 06108 studying the product architecture of a large commercial aircraft engine. We illustrate the use of these measures to test the impact of modularity on component redesign. Our results show that the relationship between component modularity and component redesign de- pends on the type of interfaces connecting product components. We also discuss direc- tions for future work. ͓DOI: 10.1115/1.2771182͔ 1 Introduction The need to measure modularity has been highlighted implicitly by Saleh ͓12͔ in his recent invitation “to contribute to the growing Previous research on product architecture has defined modular- field of flexibility in system design” ͑p. -
Complex Networks in Computer Vision
Complex networks in computer vision Pavel Voronin Shape to Network [Backes 2009], [Backes 2010a], [Backes 2010b] Weights + thresholds Feature vectors Features = characteris>cs of thresholded undirected binary networks: degree or joint degree (avg / min / max), avg path length, clustering coefficient, etc. Mul>scale Fractal Dimension Properes • Only uses distances => rotaon invariant • Weights normalized by max distance => scale invariant • Uses pixels not curve elements => robust to noise and outliers => applicable to skeletons, mul>ple contours Face recogni>on [Goncalves 2010], [Tang 2012a] Mul>ple binarizaon thresholds Texture to Network [Chalumeau 2008], [Backes 2010c], [Backes 2013] Thresholds + features Features: degree, hierarchical degree (avg / min / max) Invariance, robustness Graph structure analysis [Tang 2012b] Graph to Network Thresholds + descriptors Degree, joint degree, clustering-distance Saliency Saccade eye movements + different fixaon >me = saliency map => model them as walks in networks [Harel 2006], [Costa 2007], [Gopalakrishnan 2010], [Pal 2010], [Kim 2013] Network construc>on Nodes • pixels • segments • blocks Edges • local (neighbourhood) • global (most similar) • both Edge weights = (distance in feature space) / (distance in image space) Features: • relave intensity • entropy of local orientaons • compactness of local colour Random walks Eigenvector centrality == staonary distribu>on for Markov chain == expected >me a walker spends in the node Random walk with restart (RWR) If we have prior info on node importance,