We consider the problem of detecting communities or modules in networks, groups of vertices with a higher-than-average 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 real-world complex networks. I.

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High-throughput 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 inter-module 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 super-kingdoms. 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

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 real-world 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

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

Dense subgraphs of sparse graphs (communities), which appear in most real-world 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 real-world 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.

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 neighborhood-based 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 higher-order 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 semi-paths. We argue that this class of models holds new promise for the development of empirically plausible models for networks and network-based processes. 2 1.

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

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.

Abstract—We present a novel approach to speech processing based on the principle of pattern discovery. Our work represents a departure from traditional models of speech recognition, where the end goal is to classify speech into categories defined by a prespecified inventory of lexical units (i.e., phones or words). Instead, we attempt to discover such an inventory in an unsupervised manner by exploiting the structure of repeating patterns within the speech signal. We show how pattern discovery can be used to automatically acquire lexical entities directly from an untranscribed audio stream. Our approach to unsupervised word acquisition utilizes a segmental variant of a widely used dynamic programming technique, which allows us to find matching acoustic patterns between spoken utterances. By aggregating information about these matching patterns across audio streams, we demonstrate how to group similar acoustic sequences together to form clusters corresponding to lexical entities such as words and short multiword phrases. On a corpus of academic lecture material, we demonstrate that clusters found using this technique exhibit high purity and that many of the corresponding lexical identities are relevant to the underlying audio stream. Index Terms—Speech processing, unsupervised pattern discovery, word acquisition. I.