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Finding community structure in networks using the eigenvectors of matrices
, 2006
"... 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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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.
Comparing community structure identification
 Journal of Statistical Mechanics: Theory and Experiment
, 2005
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A measure of betweenness centrality based on random walks
 Social Networks
, 2005
"... Betweenness is a measure of the centrality of a node in a network, and is normally calculated as the fraction of shortest paths between node pairs that pass through the node of interest. Betweenness is, in some sense, a measure of the influence a node has over the spread of information through the n ..."
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Cited by 280 (0 self)
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Betweenness is a measure of the centrality of a node in a network, and is normally calculated as the fraction of shortest paths between node pairs that pass through the node of interest. Betweenness is, in some sense, a measure of the influence a node has over the spread of information through the network. By counting only shortest paths, however, the conventional definition implicitly assumes that information spreads only along those shortest paths. Here we propose a betweenness measure that relaxes this assumption, including contributions from essentially all paths between nodes, not just the shortest, although it still gives more weight to short paths. The measure is based on random walks, counting how often a node is traversed by a random walk between two other nodes. We show how our measure can be calculated using matrix methods, and give some examples of its application to particular networks. 1
Data Clustering: 50 Years Beyond KMeans
, 2008
"... Organizing data into sensible groupings is one of the most fundamental modes of understanding and learning. As an example, a common scheme of scientific classification puts organisms into taxonomic ranks: domain, kingdom, phylum, class, etc.). Cluster analysis is the formal study of algorithms and m ..."
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Cited by 274 (6 self)
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Organizing data into sensible groupings is one of the most fundamental modes of understanding and learning. As an example, a common scheme of scientific classification puts organisms into taxonomic ranks: domain, kingdom, phylum, class, etc.). Cluster analysis is the formal study of algorithms and methods for grouping, or clustering, objects according to measured or perceived intrinsic characteristics or similarity. Cluster analysis does not use category labels that tag objects with prior identifiers, i.e., class labels. The absence of category information distinguishes data clustering (unsupervised learning) from classification or discriminant analysis (supervised learning). The aim of clustering is exploratory in nature to find structure in data. Clustering has a long and rich history in a variety of scientific fields. One of the most popular and simple clustering algorithms, Kmeans, was first published in 1955. In spite of the fact that Kmeans was proposed over 50 years ago and thousands of clustering algorithms have been published since then, Kmeans is still widely used. This speaks to the difficulty of designing a general purpose clustering algorithm and the illposed problem of clustering. We provide a brief overview of clustering, summarize well known clustering methods, discuss the major challenges and key issues in designing clustering algorithms, and point out some of the emerging and useful research directions, including semisupervised clustering, ensemble clustering, simultaneous feature selection, and data clustering and large scale data clustering.
BUBBLE Rap: Socialbased forwarding in delay tolerant networks
 in Proc. ACM MobiHoc
, 2008
"... In this paper we seek to improve our understanding of human mobility in terms of social structures, and to use these structures in the design of forwarding algorithms for Pocket Switched Networks (PSNs). Taking human mobility traces from the real world, we discover that human interaction is heteroge ..."
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Cited by 269 (31 self)
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In this paper we seek to improve our understanding of human mobility in terms of social structures, and to use these structures in the design of forwarding algorithms for Pocket Switched Networks (PSNs). Taking human mobility traces from the real world, we discover that human interaction is heterogeneous both in terms of hubs (popular individuals) and groups or communities. We propose a social based forwarding algorithm, BUBBLE, which is shown empirically to improve the forwarding efficiency significantly compared to oblivious forwarding schemes and to PROPHET algorithm. We also show how this algorithm can be implemented in a distributed way, which demonstrates that it is applicable in the decentralised environment of PSNs.
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 256 (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
Statistical properties of community structure in large social and information networks
"... A large body of work has been devoted to identifying community structure in networks. A community is often though of as a set of nodes that has more connections between its members than to the remainder of the network. In this paper, we characterize as a function of size the statistical and structur ..."
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Cited by 242 (14 self)
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A large body of work has been devoted to identifying community structure in networks. A community is often though of as a set of nodes that has more connections between its members than to the remainder of the network. In this paper, we characterize as a function of size the statistical and structural properties of such sets of nodes. We define the network community profile plot, which characterizes the “best ” possible community—according to the conductance measure—over a wide range of size scales, and we study over 70 large sparse realworld networks taken from a wide range of application domains. Our results suggest a significantly more refined picture of community structure in large realworld networks than has been appreciated previously. Our most striking finding is that in nearly every network dataset we examined, we observe tight but almost trivial communities at very small scales, and at larger size scales, the best possible communities gradually “blend in ” with the rest of the network and thus become less “communitylike.” This behavior is not explained, even at a qualitative level, by any of the commonlyused network generation models. Moreover, this behavior is exactly the opposite of what one would expect based on experience with and intuition from expander graphs, from graphs that are wellembeddable in a lowdimensional structure, and from small social networks that have served as testbeds of community detection algorithms. We have found, however, that a generative model, in which new edges are added via an iterative “forest fire” burning process, is able to produce graphs exhibiting a network community structure similar to our observations.
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 224 (3 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.
Community structure in large networks: Natural cluster sizes and the absence of large welldefined clusters
, 2008
"... A large body of work has been devoted to defining and identifying clusters or communities in social and information networks, i.e., in graphs in which the nodes represent underlying social entities and the edges represent some sort of interaction between pairs of nodes. Most such research begins wit ..."
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Cited by 198 (17 self)
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A large body of work has been devoted to defining and identifying clusters or communities in social and information networks, i.e., in graphs in which the nodes represent underlying social entities and the edges represent some sort of interaction between pairs of nodes. Most such research begins with the premise that a community or a cluster should be thought of as a set of nodes that has more and/or better connections between its members than to the remainder of the network. In this paper, we explore from a novel perspective several questions related to identifying meaningful communities in large social and information networks, and we come to several striking conclusions. Rather than defining a procedure to extract sets of nodes from a graph and then attempt to interpret these sets as a “real ” communities, we employ approximation algorithms for the graph partitioning problem to characterize as a function of size the statistical and structural properties of partitions of graphs that could plausibly be interpreted as communities. In particular, we define the network community profile plot, which characterizes the “best ” possible community—according to the conductance measure—over a wide range of size scales. We study over 100 large realworld networks, ranging from traditional and online social networks, to technological and information networks and
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 166 (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