To find interesting structure in networks, community detection algorithms have to take into account not only the network topology, but also dynamics of interactions between nodes. We investigate this claim using the paradigm of synchronization in a network of coupled oscillators. As the network evolves to a global steady state, nodes belonging to the same community synchronize faster than nodes belonging to different communities. Traditionally, nodes in network synchronization models are coupled via one-to-one, or conservative interactions. However, social interactions are often one-to-many, as for example, in social media, where users broadcast messages to all their followers. We formulate a novel model of synchronization in a network of coupled oscillators in which the oscillators are coupled via one-to-many, or non-conservative interactions. We study the dynamics of different interaction models and contrast their spectral properties. To find multi-scale community structure in a network of interacting nodes, we define a similarity function that measures the degree to which nodes are synchronized and use it to hierarchically cluster nodes. We study real-world social networks, including networks of two social media providers. To evaluate the quality of the discovered communities in a social media network we propose a community quality metric based on user activity. We find that conservative and non-conservative interaction models lead to dramatically different views of community structure even within the same network. Our work offers a novel mathematical framework for exploring the relationship between network structure, topology and dynamics.
Community detection is an extremely active research area, with a variety of methods proposed, including spectral clustering, graph partitioning and modularity maximization @cite_4 . We show in this paper that these methods can be expressed in terms of the generalized linear interaction model that assumes a specific type of interaction. We also demonstrate that to get the full picture of network's emergent structure, community detection method must account for the dynamic process occurring on the network.
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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. Most methods proposed to uncover communities in complex networks rely on combinatorial graph properties. Usually an edge-counting quality function, such as modularity, is optimized over all partitions of the graph compared against a null random graph model. Here we introduce a systematic dynamical framework to design and analyze a wide variety of quality functions for community detection. The quality of a partition is measured by its Markov Stability, a time-parametrized function defined in terms of the statistical properties of a Markov process taking place on the graph. The Markov process provides a dynamical sweeping across all scales in the graph, and the time scale is an intrinsic parameter that uncovers communities at different resolutions. This dynamic-based community detection leads to a compound optimization, which favours communities of comparable centrality (as defined by the stationary distribution), and provides a unifying framework for spectral algorithms, as well as different heuristics for community detection, including versions of modularity and Potts model. Our dynamic framework creates a systematic link between different stochastic dynamics and their corresponding notions of optimal communities under distinct (node and edge) centralities. We show that the Markov Stability can be computed efficiently to find multi-scale community structure in large networks. We consider three distinct and well-studied problems concerning network structure: community detection by modularity maximization, community detection by statistical inference, and normalized-cut graph partitioning. Each of these problems can be tackled using spectral algorithms that make use of the eigenvectors of matrix representations of the network. We show that with certain choices of the free parameters appearing in these spectral algorithms the algorithms for all three problems are, in fact, identical, and hence that, at least within the spectral approximations used here, there is no difference between the modularity- and inference-based community detection methods, or between either and graph partitioning. Most methods proposed to uncover communities in complex networks rely on their structural properties. Here we introduce the stability of a network partition, a measure of its quality defined in terms of the statistical properties of a dynamical process taking place on the graph. The time-scale of the process acts as an intrinsic parameter that uncovers community structures at different resolutions. The stability extends and unifies standard notions for community detection: modularity and spectral partitioning can be seen as limiting cases of our dynamic measure. Similarly, recently proposed multi-resolution methods correspond to linearisations of the stability at short times. The connection between community detection and Laplacian dynamics enables us to establish dynamically motivated stability measures linked to distinct null models. We apply our method to find multi-scale partitions for different networks and show that the stability can be computed efficiently for large networks with extended versions of current algorithms. Community structure is one of the main structural features of networks, revealing both their internal organization and the similarity of their elementary units. Despite the large variety of methods proposed to detect communities in graphs, there is a big need for multi-purpose techniques, able to handle different types of datasets and the subtleties of community structure. In this paper we present OSLOM (Order Statistics Local Optimization Method), the first method capable to detect clusters in networks accounting for edge directions, edge weights, overlapping communities, hierarchies and community dynamics. It is based on the local optimization of a fitness function expressing the statistical significance of clusters with respect to random fluctuations, which is estimated with tools of Extreme and Order Statistics. OSLOM can be used alone or as a refinement procedure of partitions covers delivered by other techniques. We have also implemented sequential algorithms combining OSLOM with other fast techniques, so that the community structure of very large networks can be uncovered. Our method has a comparable performance as the best existing algorithms on artificial benchmark graphs. Several applications on real networks are shown as well. OSLOM is implemented in a freely available software (http: www.oslom.org), and we believe it will be a valuable tool in the analysis of networks. An efficient and relatively fast algorithm for the detection of communities in complex networks is introduced. The method exploits spectral properties of the graph Laplacian matrix combined with hierarchical clustering techniques, and includes a procedure for maximizing the 'modularity' of the output. Its performance is compared with that of other existing methods, as applied to different well-known instances of complex networks with a community structure, both computer generated and from the real world. Our results are, in all the cases tested, at least as good as the best ones obtained with any other methods, and faster in most of the cases than methods providing similar quality results. This converts the algorithm into a valuable computational tool for detecting and analysing communities and modular structures in complex networks. Graph clustering has been widely applied in exploring regularities emerging in relational data. Recently, the rapid development of network theory correlates graph clustering with the detection of community structure, a common and important topological characteristic of networks. Most existing methods investigate the community structure at a single topological scale. However, as shown by empirical studies, the community structure of real world networks often exhibits multiple topological descriptions, corresponding to the clustering at different resolutions. Furthermore, the detection of multiscale community structure is heavily affected by the heterogeneous distribution of node degree. It is very challenging to detect multiscale community structure in heterogeneous networks. In this paper, we propose a novel, unified framework for detecting community structure from the perspective of dimensionality reduction. Based on the framework, we first prove that the well-known Laplacian matrix for network partition and the widely-used modularity matrix for community detection are two kinds of covariance matrices used in dimensionality reduction. We then propose a novel method to detect communities at multiple topological scales within our framework. We further show that existing algorithms fail to deal with heterogeneous node degrees. We develop a novel method to handle heterogeneity of networks by introducing a rescaling transformation into the covariance matrices in our framework. Extensive tests on real world and artificial networks demonstrate that the proposed correlation matrices significantly outperform Laplacian and modularity matrices in terms of their ability to identify multiscale community structure in heterogeneous networks. Many networks of interest in the sciences, including social networks, computer networks, and metabolic and regulatory networks, are found to divide naturally into communities or modules. The problem of detecting and characterizing this community structure is one of the outstanding issues in the study of networked systems. One highly effective approach is the optimization of the quality function known as “modularity” over the possible divisions of a network. Here I show that the modularity can be expressed in terms of the eigenvectors of a characteristic matrix for the network, which I call the modularity matrix, and that this expression leads to a spectral algorithm for community detection that returns results of demonstrably higher quality than competing methods in shorter running times. I illustrate the method with applications to several published network data sets. Many researchers have studied complex networks such as the World Wide Web, social networks, and the protein interaction network. They have found scale-free characteristics, the small-world effect, the property of high-clustering coefficient, and so on. One hot topic in this area is community detection. For example, the community shows a set of web pages about a certain topic in the WWW. The community structure is unquestionably a key characteristic of complex networks. In this paper, we propose a new method for finding communities in complex networks. Our proposed method considers the overlaps between communities using the concept of the intersection graph. Additionally, we address the problem of edge in homogeneity by weighting edges using the degree of overlaps and the similarity of content information between sets. Finally, we conduct clustering based on modularity. And then, we evaluate our method on a real SNS network. With the recent advances in information networks, the problem of identifying group structure or communities has received a significant amount of attention. Most of the existing principles of community detection or clustering mainly focus on either the topological structure of a network or the node attributes separately, while both of the two aspects provide valuable information to characterize the nature of communities. In this paper we combine the topological structure of a network as well as the content information of nodes in the task of detecting communities in information networks. Specifically, we treat a network as a dynamic system and consider its community structure as a consequence of interactions among nodes. To model the interactions we introduce the principle of content propagation and integrate the aspects of structure and content in a network naturally. We further describe the interactions among nodes in two different ways, including a linear model to approximate influence propagation, and modeling the interactions directly with random walk. Based on interaction modeling, the nature of communities is described by analyzing the stable status of the dynamic system. Extensive experimental results on benchmark datasets demonstrate the superiority of the proposed framework over the state of the art. We study networks that display community structure -- groups of nodes within which connections are unusually dense. Using methods from random matrix theory, we calculate the spectra of such networks in the limit of large size, and hence demonstrate the presence of a phase transition in matrix methods for community detection, such as the popular modularity maximization method. The transition separates a regime in which such methods successfully detect the community structure from one in which the structure is present but is not detected. By comparing these results with recent analyses of maximum-likelihood methods we are able to show that spectral modularity maximization is an optimal detection method in the sense that no other method will succeed in the regime where the modularity method fails. Communities are fundamental entities for the characterization of the structure of real networks. The standard approachtotheidentificationofcommunitiesinnetworksisbasedontheoptimizationofaqualityfunctionknown as modularity. Although modularity has been at the center of an intense research activity and many methods for its maximization have been proposed, not much is yet known about the necessary conditions that communities need to satisfy in order to be detectable with modularity maximization methods. Here, we develop a simple theory to establish these conditions, and we successfully apply it to various classes of network models. Our main result is that heterogeneity in the degree distribution helps modularity to correctly recover the community structure of a network and that, in the realistic case of scale-free networks with degree exponent γ< 2.5, modularity is always able to detect the presence of communities. Many real-world networks are so large that we must simplify their structure before we can extract useful information about the systems they represent. As the tools for doing these simplifications proliferate within the network literature, researchers would benefit from some guidelines about which of the so-called community detection algorithms are most appropriate for the structures they are studying and the questions they are asking. Here we show that different methods highlight different aspects of a network's structure and that the the sort of information that we seek to extract about the system must guide us in our decision. For example, many community detection algorithms, including the popular modularity maximization approach, infer module assignments from an underlying model of the network formation process. However, we are not always as interested in how a system's network structure was formed, as we are in how a network's extant structure influences the system's behavior. To see how structure influences current behavior, we will recognize that links in a network induce movement across the network and result in system-wide interdependence. In doing so, we explicitly acknowledge that most networks carry flow. To highlight and simplify the network structure with respect to this flow, we use the map equation. We present an intuitive derivation of this flow-based and information-theoretic method and provide an interactive on-line application that anyone can use to explore the mechanics of the map equation. The differences between the map equation and the modularity maximization approach are not merely conceptual. Because the map equation attends to patterns of flow on the network and the modularity maximization approach does not, the two methods can yield dramatically different results for some network structures. To illustrate this and build our understanding of each method, we partition several sample networks. We also describe an algorithm and provide source code to efficiently decompose large weighted and directed networks based on the map equation. An increasing number of users interact, collaborate, and share information through social networks. Unprecedented growth in social networks is generating a significant amount of unstructured social data. From such data, distilling communities where users have common interests and tracking variations of users' interests over time are important research tracks in fields such as opinion mining, trend prediction, and personalized services. However, these tasks are extremely difficult considering the highly dynamic characteristics of the data. Existing community detection methods are time consuming, making it difficult to process data in real time. In this paper, dynamic unstructured data is modeled as a stream. Tag assignments stream clustering (TASC), an incremental scalable community detection method, is proposed based on locality-sensitive hashing. Both tags and latent interactions among users are incorporated in the method. In our experiments, the social dynamic behaviors of users are first analyzed. The proposed TASC method is then compared with state-of-the-art clustering methods such as StreamKmeans and incremental k-clique; results indicate that TASC can detect communities more efficiently and effectively. Dynamics of social systems are the result of the complex superposition of interactions taking place at different scales, ranging from the pairwise communications between individuals to the macroscopic evolutionary patterns of the full interaction graph. Social communities, namely groups of people originated by any spontaneous aggregation process, constitute the mid-ground between such two extremes. Groups are important constituents of social environments as they form the basis for people’s participation and engagement beyond their minute dyadic interactions. Communities in online social media have been studied widely in their static and evolutionary aspects, but only recently some attention has been devoted to the exploration of their nature. Besides the characterization of online communities along their spatio-temporal and activity features, the recent advancements in the emerging field of computational sociology have provided a new lens to study social aggregations along their social and topical dimensions. Using the online photo sharing community Flickr as a main running example, we survey some techniques that have been used to get a multi-faceted description of group types and we show that different types of groups impact on orthogonal interaction processes on the social graph, such as the diffusion of information along social ties. Our overview supports the intuition that a more nuanced description of groups could not only improve the understanding of the activity of the user base but can also foster a better interpretation of other phenomena occurring on social graphs. The problem of decomposing networks into modules (or clusters) has gained much attention in recent years, as it can account for a coarse-grained description of complex systems, often revealing functional subunits of these systems. A variety of module detection algorithms have been proposed, mostly oriented towards finding hard partitionings of undirected networks. Despite the increasing number of fuzzy clustering methods for directed networks, many of these approaches tend to neglect important directional information. In this paper, we present a novel random walk based approach for finding fuzzy partitions of directed, weighted networks, where edge directions play a crucial role in defining how well nodes in a module are interconnected. We will show that cycle decomposition of a random walk process connects the notion of network modules and information transport in a network, leading to a new, symmetric measure of node communication. Finally, we will use this measure to introduce a communication graph, for which we will show that although being undirected it inherits important directional information of modular structures from the original network. We demonstrate an exact equivalence between two widely used methods of community detection in networks, the method of modularity maximization in its generalized form which incorporates a resolution parameter controlling the size of the communities discovered, and the method of maximum likelihood applied to the special case of the stochastic block model known as the planted partition model, in which all communities in a network are assumed to have statistically similar properties. Among other things, this equivalence provides a mathematically principled derivation of the modularity function, clarifies the conditions and assumptions of its use, and gives an explicit formula for the optimal value of the resolution parameter. Community detection in signed complex networks is a challenging research problem aiming at finding groups of entities having positive connections within the same cluster and negative relationships between different clusters. Most of the proposed approaches have been developed for networks having only positive edges. In this paper we propose a multiobjective approach to detect communities in signed networks. The method partitions a network in groups of nodes such that two objectives are contemporarily optimized. The former is that the partitioning should have dense positive intra-connections and sparse negative interconnections, the latter is that it should have as few as possible negative intra-connections and positive inter-connections. We show that the concepts of signed modularity and frustration fulfill these objectives, and that the maximization of signed modularity and the minimization of frustration allow to obtain very good solutions to the problem. An extensive set of experiments on both real-life and synthetic signed networks shows the efficacy of the approach. It has been shown that the communities of complex networks often overlap with each other. However, there is no effective method to quantify the overlapping community structure. In this paper, we propose a metric to address this problem. Instead of assuming that one node can only belong to one community, our metric assumes that a maximal clique only belongs to one community. In this way, the overlaps between communities are allowed. To identify the overlapping community structure, we construct a maximal clique network from the original network, and prove that the optimization of our metric on the original network is equivalent to the optimization of Newman's modularity on the maximal clique network. Thus the overlapping community structure can be identified through partitioning the maximal clique network using any modularity optimization method. The effectiveness of our metric is demonstrated by extensive tests on both artificial networks and real world networks with a known community structure. The application to the word association network also reproduces excellent results. Community discovery has emerged during the last decade as one of the most challenging problems in social network analysis. Many algorithms have been proposed to find communities on static networks, i.e. networks which do not change in time. However, social networks are dynamic realities (e.g. call graphs, online social networks): in such scenarios static community discovery fails to identify a partition of the graph that is semantically consistent with the temporal information expressed by the data. In this work we propose Tiles, an algorithm that extracts overlapping communities and tracks their evolution in time following an online iterative procedure. Our algorithm operates following a domino effect strategy, dynamically recomputing nodes community memberships whenever a new interaction takes place. We compare Tiles with state-of-the-art community detection algorithms on both synthetic and real world networks having annotated community structure: our experiments show that the proposed approach is able to guarantee lower execution times and better correspondence with the ground truth communities than its competitors. Moreover, we illustrate the specifics of the proposed approach by discussing the properties of identified communities it is able to identify.