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75
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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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
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.
Graph theory and networks in biology
 IET Systems Biology, 1:89 – 119
, 2007
"... In this paper, we present a survey of the use of graph theoretical techniques in Biology. In particular, we discuss recent work on identifying and modelling the structure of biomolecular networks, as well as the application of centrality measures to interaction networks and research on the hierarch ..."
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In this paper, we present a survey of the use of graph theoretical techniques in Biology. In particular, we discuss recent work on identifying and modelling the structure of biomolecular networks, as well as the application of centrality measures to interaction networks and research on the hierarchical structure of such networks and network motifs. Work on the link between structural network properties and dynamics is also described, with emphasis on synchronization and disease propagation. 1
Community Structure in Graphs
, 2007
"... Graph vertices are often organized into groups that seem to live fairly independently of the rest of the graph, with which they share but a few edges, whereas the relationships between group members are stronger, as shown by the large number of mutual connections. Such groups of vertices, or communi ..."
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Cited by 44 (0 self)
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Graph vertices are often organized into groups that seem to live fairly independently of the rest of the graph, with which they share but a few edges, whereas the relationships between group members are stronger, as shown by the large number of mutual connections. Such groups of vertices, or communities, can be considered as independent compartments of a graph. Detecting communities is of great importance in sociology, biology and computer science, disciplines where systems are often represented as graphs. The task is very hard, though, both conceptually, due to the ambiguity in the definition of community and in the discrimination of different partitions and practically, because algorithms must find “good ” partitions among an exponentially large number of them. Other complications are represented by the possible occurrence of hierarchies, i.e. communities which are nested inside larger communities, and by the existence of overlaps between communities, due to the presence of nodes belonging to more groups. All these aspects are dealt with in some detail and many methods are described, from traditional approaches used in computer science and sociology to recent techniques developed mostly within statistical physics.
Multilevel algorithms for modularity clustering
"... been adapted to modularity clustering. Section 4 details the singlelevel and multilevel refinement heuristics, and Section 5.3 compares them experimentally. Because the effectiveness of (particularly multilevel) refinement may depend on the coarsening algorithm, Section 5.4 examines various combi ..."
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been adapted to modularity clustering. Section 4 details the singlelevel and multilevel refinement heuristics, and Section 5.3 compares them experimentally. Because the effectiveness of (particularly multilevel) refinement may depend on the coarsening algorithm, Section 5.4 examines various combinations of coarsening and refinement heuristics. Section 6 compares public implementations and benchmark results of modularity clustering heuristics, without a restriction to coarsening and refinement algorithms. While this is one of the most extensive comparisons in the literature, it is far from exhaustive, because implementations and sufficient experimental results have not been published for some proposed heurisarXiv:0812.4073v1
Detecting highly overlapping community structure by greedy clique expansion,” KDD SNA
, 2010
"... In complex networks it is common for each node to belong to several communities, implying a highly overlapping community structure. Recent advances in benchmarking indicate that the existing community assignment algorithms that are capable of detecting overlapping communities perform well only w ..."
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Cited by 32 (6 self)
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In complex networks it is common for each node to belong to several communities, implying a highly overlapping community structure. Recent advances in benchmarking indicate that the existing community assignment algorithms that are capable of detecting overlapping communities perform well only when the extent of community overlap is kept to modest levels. To overcome this limitation, we introduce a new community assignment algorithm called Greedy Clique Expansion (GCE). The algorithm identifies distinct cliques as seeds and expands these seeds by greedily optimizing a local fitness function. We perform extensive benchmarks on synthetic data to demonstrate that GCE’s good performance is robust across diverse graph topologies. Significantly, GCE is the only algorithm to perform well on these synthetic graphs, in which every node belongs to multiple communities. Furthermore, when put to the task of identifying functional modules in protein interaction data, and college dorm assignments in Facebook friendship data, we find that GCE performs competitively.
An experimental investigation of graph kernels on a collaborative recommendation task
 Proceedings of the 6th International Conference on Data Mining (ICDM 2006
, 2006
"... This paper presents a survey as well as a systematic empirical comparison of seven graph kernels and two related similarity matrices (simply referred to as graph kernels), namely the exponential diffusion kernel, the Laplacian exponential diffusion kernel, the von Neumann diffusion kernel, the regul ..."
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Cited by 27 (7 self)
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This paper presents a survey as well as a systematic empirical comparison of seven graph kernels and two related similarity matrices (simply referred to as graph kernels), namely the exponential diffusion kernel, the Laplacian exponential diffusion kernel, the von Neumann diffusion kernel, the regularized Laplacian kernel, the commutetime kernel, the randomwalkwithrestart similarity matrix, and finally, three graph kernels introduced in this paper: the regularized commutetime kernel, the Markov diffusion kernel, and the crossentropy diffusion matrix. The kernelonagraph approach is simple and intuitive. It is illustrated by applying the nine graph kernels to a collaborativerecommendation task and to a semisupervised classification task, both on several databases. The graph methods compute proximity measures between nodes that help study the structure of the graph. Our comparisons suggest that the regularized commutetime and the Markov diffusion kernels perform best, closely followed by the regularized Laplacian kernel. 1
Graph nodes clustering with the sigmoid commutetime kernel: A . . .
 DATA & KNOWLEDGE ENGINEERING
, 2009
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