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273
Finding community structure in networks using the eigenvectors of matrices. Phys
 Rev. E
"... We consider the problem of detecting communities or modules in networks, groups of vertices with a higherthanaverage density of edges connecting them. Previous work indicates that a robust approach to this problem is the maximization of the benefit function known as “modularity ” over possible div ..."
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Cited by 232 (0 self)
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We consider the problem of detecting communities or modules in networks, groups of vertices with a higherthanaverage density of edges connecting them. Previous work indicates that a robust approach to this problem is the maximization of the benefit function known as “modularity ” over possible divisions of a network. Here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix, which plays a role in community detection similar to that played by the graph Laplacian in graph partitioning calculations. This result leads us to a number of possible algorithms for detecting community structure, as well as several other results, including a spectral measure of bipartite structure in networks and a new centrality measure that identifies those vertices that occupy central positions within the communities to which they belong. The algorithms and measures proposed are illustrated with applications to a variety of realworld complex networks. I.
Comparing community structure identification
 Journal of Statistical Mechanics: Theory and Experiment
, 2005
"... ..."
Nunes Amaral. Functional cartography of complex metabolic networks
 Nature
, 2005
"... Highthroughput techniques are leading to an explosive growth in the size of biological databases and creating the opportunity to revolutionize our understanding of life and disease. Interpretation of these data remains, however, a major scientific challenge. Here, we propose a methodology that enab ..."
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Cited by 132 (3 self)
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Highthroughput techniques are leading to an explosive growth in the size of biological databases and creating the opportunity to revolutionize our understanding of life and disease. Interpretation of these data remains, however, a major scientific challenge. Here, we propose a methodology that enables us to extract and display information contained in complex networks 1,2,3. Specifically, we demonstrate that one can (i) find functional modules 4,5 in complex networks, and (ii) classify nodes into universal roles according to their pattern of intra and intermodule connections. The method thus yields a “cartographic representation ” of complex networks. Metabolic networks 6,7,8 are among the most challenging biological networks and, arguably, the ones with more potential for immediate applicability 9. We use our method to analyze the metabolic networks of twelve organisms from three different superkingdoms. We find that, typically, 80 % of the nodes are only connected to other nodes within their respective modules, and that nodes with different roles are affected by different evolutionary constraints and pressures. Remarkably, we
A spectral clustering approach to finding communities in graphs
 In SIAM International Conference on Data Mining
, 2005
"... Clustering nodes in a graph is a useful general technique in data mining of large network data sets. In this context, Newman and Girvan [9] recently proposed an objective function for graph clustering called the Q function which allows automatic selection of the number of clusters. Empirically, high ..."
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Cited by 103 (0 self)
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Clustering nodes in a graph is a useful general technique in data mining of large network data sets. In this context, Newman and Girvan [9] recently proposed an objective function for graph clustering called the Q function which allows automatic selection of the number of clusters. Empirically, higher values of the Q function have been shown to correlate well with good graph clusterings. In this paper we show how optimizing the Q function can be reformulated as a spectral relaxation problem and propose two new spectral clustering algorithms that seek to maximize Q. Experimental results indicate that the new algorithms are efficient and effective at finding both good clusterings and the appropriate number of clusters across a variety of realworld graph data sets. In addition, the spectral algorithms are much faster for large sparse graphs, scaling roughly linearly with the number of nodes n in the graph, compared to O(n 2) for previous clustering algorithms using the Q function. 1
Computing communities in large networks using random walks
 J. of Graph Alg. and App. bf
, 2004
"... Dense subgraphs of sparse graphs (communities), which appear in most realworld complex networks, play an important role in many contexts. Computing them however is generally expensive. We propose here a measure of similarities between vertices based on random walks which has several important advan ..."
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Cited by 98 (2 self)
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Dense subgraphs of sparse graphs (communities), which appear in most realworld complex networks, play an important role in many contexts. Computing them however is generally expensive. We propose here a measure of similarities between vertices based on random walks which has several important advantages: it captures well the community structure in a network, it can be computed efficiently, and it can be used in an agglomerative algorithm to compute efficiently the community structure of a network. We propose such an algorithm, called Walktrap, which runs in time O(mn 2) and space O(n 2) in the worst case, and in time O(n 2 log n) and space O(n 2) in most realworld cases (n and m are respectively the number of vertices and edges in the input graph). Extensive comparison tests show that our algorithm surpasses previously proposed ones concerning the quality of the obtained community structures and that it stands among the best ones concerning the running time.
Characterization of complex networks: A survey of measurements
 Advances in Physics
"... Each complex network (or class of networks) presents specific topological features which characterize its connectivity and highly influence the dynamics and function of processes executed on the network. The analysis, discrimination, and synthesis of complex networks therefore rely on the use of mea ..."
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Cited by 89 (7 self)
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Each complex network (or class of networks) presents specific topological features which characterize its connectivity and highly influence the dynamics and function of processes executed on the network. The analysis, discrimination, and synthesis of complex networks therefore rely on the use of measurements capable of expressing the most relevant topological features. This article presents a survey of such measurements. It includes general considerations about complex network characterization, a brief review of the principal models, and the presentation of the main existing measurements organized into classes. Special attention is given to relating complex network analysis with the areas of pattern recognition and feature selection, as well as on surveying some concepts and measurements from traditional graph theory which are potentially useful for complex network research. Depending on the network and the analysis task one has in mind, a specific set of features may be chosen. It is hoped that the present survey will help the
On Modularity Clustering
, 2008
"... Modularity is a recently introduced quality measure for graph clusterings. It has immediately received considerable attention in several disciplines, and in particular in the complex systems literature, although its properties are not well understood. We study the problem of finding clusterings wit ..."
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Cited by 70 (12 self)
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Modularity is a recently introduced quality measure for graph clusterings. It has immediately received considerable attention in several disciplines, and in particular in the complex systems literature, although its properties are not well understood. We study the problem of finding clusterings with maximum modularity, thus providing theoretical foundations for past and present work based on this measure. More precisely, we prove the conjectured hardness of maximizing modularity both in the general case and with the restriction to cuts, and give an Integer Linear Programming formulation. This is complemented by first insights into the behavior and performance of the commonly applied greedy agglomerative approach.
Balancing Systematic and Flexible Exploration of Social Networks
 IEEE Transactions on Visualization and Computer Graphics
, 2006
"... Abstract — Social network analysis (SNA) has emerged as a powerful method for understanding the importance of relationships in networks. However, interactive exploration of networks is currently challenging because: (1) it is difficult to find patterns and comprehend the structure of networks with m ..."
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Cited by 69 (21 self)
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Abstract — Social network analysis (SNA) has emerged as a powerful method for understanding the importance of relationships in networks. However, interactive exploration of networks is currently challenging because: (1) it is difficult to find patterns and comprehend the structure of networks with many nodes and links, and (2) current systems are often a medley of statistical methods and overwhelming visual output which leaves many analysts uncertain about how to explore in an orderly manner. This results in exploration that is largely opportunistic. Our contributions are techniques to help structural analysts understand social networks more effectively. We present SocialAction, a system that uses attribute ranking and coordinated views to help users systematically examine numerous SNA measures. Users can (1) flexibly iterate through visualizations of measures to gain an overview, filter nodes, and find outliers, (2) aggregate networks using link structure, find cohesive subgroups, and focus on communities of interest, and (3) untangle networks by viewing different link types separately, or find patterns across different link types using a matrix overview. For each operation, a stable node layout is maintained in the network visualization so users can make comparisons. SocialAction offers analysts a strategy beyond opportunism, as it provides systematic, yet flexible, techniques for exploring social networks. Index Terms—Social networks, interactive graph visualization, attribute ranking, coordinated views, exploratory data analysis. 1
NeighborhoodBased Models for Social Networks
 Sociological Methodology
, 2002
"... Harrison White and several anonymous reviewers for valuable comments on the work. We argue that social networks can be modeled as the outcome of processes that occur in overlapping local regions of the network, termed local social neighborhoods. Each neighborhood is conceived as a possible site of i ..."
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Cited by 55 (4 self)
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Harrison White and several anonymous reviewers for valuable comments on the work. We argue that social networks can be modeled as the outcome of processes that occur in overlapping local regions of the network, termed local social neighborhoods. Each neighborhood is conceived as a possible site of interaction and corresponds to a subset of possible network ties. In this paper, we discuss hypotheses about the form of these neighborhoods, and we present two new and theoretically plausible ways in which neighborhoodbased models for networks can be constructed. In the first, we introduce the notion of a setting structure, a directly hypothesized (or observed) set of exogenous constraints on possible neighborhood forms. In the second, we propose higherorder neighborhoods that are generated, in part, by the outcome of interactive network processes themselves. Applications of both approaches to model construction are presented, and the developments are considered within a general conceptual framework of locale for social networks. We show how assumptions about neighborhoods can be cast within a hierarchy of increasingly complex models; these models represent a progressively greater capacity for network processes to “reach ” across a network through long cycles or semipaths. We argue that this class of models holds new promise for the development of empirically plausible models for networks and networkbased processes. 2 1.
An Analysis of Social NetworkBased Sybil Defenses ABSTRACT
"... Recently, there has been much excitement in the research community over using social networks to mitigate multiple identity, or Sybil, attacks. A number of schemes have been proposed, but they differ greatly in the algorithms they use and in the networks upon which they are evaluated. As a result, t ..."
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Cited by 47 (6 self)
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Recently, there has been much excitement in the research community over using social networks to mitigate multiple identity, or Sybil, attacks. A number of schemes have been proposed, but they differ greatly in the algorithms they use and in the networks upon which they are evaluated. As a result, the research community lacks a clear understanding of how these schemes compare against each other, how well they would work on realworld social networks with different structural properties, or whether there exist other (potentially better) ways of Sybil defense. In this paper, we show that, despite their considerable differences, existing Sybil defense schemes work by detecting local communities (i.e., clusters of nodes more tightly knit than the rest of the graph) around a trusted node. Our finding has important implications for both existing and future designs of Sybil defense schemes. First, we show that there is an opportunity to leverage the substantial amount of prior work on general community detection algorithms in order to defend against Sybils. Second, our analysis reveals the fundamental limits of current social networkbased Sybil defenses: We demonstrate that networks with welldefined community structure are inherently more vulnerable to Sybil attacks, and that, in such networks, Sybils can carefully target their links in order make their attacks more effective.