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.
@cite_4
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Community detection is the task of detecting hidden communities from observed interactions. Guaranteed community detection has so far been mostly limited to models with non-overlapping communities such as the stochastic block model. In this paper, we remove this restriction, and provide guaranteed community detection for a family of probabilistic network models with overlapping communities, termed as the mixed membership Dirichlet model, first introduced by (2008). This model allows for nodes to have fractional memberships in multiple communities and assumes that the community memberships are drawn from a Dirichlet distribution. Moreover, it contains the stochastic block model as a special case. We propose a unified approach to learning these models via a tensor spectral decomposition method. Our estimator is based on low-order moment tensor of the observed network, consisting of 3-star counts. Our learning method is fast and is based on simple linear algebraic operations, e.g., singular value decomposition and tensor power iterations. We provide guaranteed recovery of community memberships and model parameters and present a careful finite sample analysis of our learning method. As an important special case, our results match the best known scaling requirements for the (homogeneous) stochastic block model. 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. Community detection is a topic of interest in the study of complex networks such as the protein–protein interaction networks and metabolic networks. In recent years, various methods were proposed to detect community structures of the networks. Here, a kind of local modularity with tunable parameter is derived from the Newman–Girvan modularity by a special self-loop strategy that depends on the community division of the networks. By the self-loop strategy, one can easily control the definition of modularity, and the resulting modularity can be optimized by using the existing modularity optimization algorithms. The local modularity is used as the target function for community detection, and a self-consistent method is proposed for the optimization of the local modularity. We analyze the behaviors of the local modularity and show the validity of the local modularity in detecting community structures on various networks. Detecting hidden communities from observed interactions is a classical problem. Theoretical analysis of community detection has so far been mostly limited to models with non-overlapping communities such as the stochastic block model. In this paper, we provide guaranteed community detection for a family of probabilistic network models with overlapping communities, termed as the mixed membership Dirichlet model, first introduced in (2008). This model allows for nodes to have fractional memberships in multiple communities and assumes that the community memberships are drawn from a Dirichlet distribution. Moreover, it contains the stochastic block model as a special case. We propose a unified approach to learning communities in these models via a tensor spectral decomposition approach. Our estimator uses low-order moment tensor of the observed network, consisting of 3-star counts. Our learning method is based on simple linear algebraic operations such as singular value decomposition and tensor power iterations. We provide guaranteed recovery of community memberships and model parameters, and present a careful finite sample analysis of our learning method. Additionally, our results match the best known scaling requirements for the special case of the (homogeneous) stochastic block model. Community detection is an important task for mining the structure and function of complex networks. Generally, there are several different kinds of nodes in a network which are cluster nodes densely connected within communities, as well as some special nodes like hubs bridging multiple communities and outliers marginally connected with a community. In addition, it has been shown that there is a hierarchical structure in complex networks with communities embedded within other communities. Therefore, a good algorithm is desirable to be able to not only detect hierarchical communities, but also identify hubs and outliers. In this paper, we propose a parameter-free hierarchical network clustering algorithm SHRINK by combining the advantages of density-based clustering and modularity optimization methods. Based on the structural connectivity information, the proposed algorithm can effectively reveal the embedded hierarchical community structure with multiresolution in large-scale weighted undirected networks, and identify hubs and outliers as well. Moreover, it overcomes the sensitive threshold problem of density-based clustering algorithms and the resolution limit possessed by other modularity-based methods. To illustrate our methodology, we conduct experiments with both real-world and synthetic datasets for community detection, and compare with many other baseline methods. Experimental results demonstrate that SHRINK achieves the best performance with consistent improvements. 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. 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 detection is a fundamental problem in network analysis which is made more challenging by overlaps between communities which often occur in practice. Here we propose a general, flexible, and interpretable generative model for overlapping communities, which can be thought of as a generalization of the degree-corrected stochastic block model. We develop an efficient spectral algorithm for estimating the community memberships, which deals with the overlaps by employing the K-medians algorithm rather than the usual K-means for clustering in the spectral domain. We show that the algorithm is asymptotically consistent when networks are not too sparse and the overlaps between communities not too large. Numerical experiments on both simulated networks and many real social networks demonstrate that our method performs very well compared to a number of benchmark methods for overlapping community detection. 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. Identification of module or community structures is important for characterizing and understanding complex systems. While designed with different objectives, i.e., stochastic models for regeneration and modularity maximization models for discrimination, both these two types of model look for low-rank embedding to best represent and reconstruct network topology. However, the mapping through such embedding is linear, whereas real networks have various nonlinear features, making these models less effective in practice. Inspired by the strong representation power of deep neural networks, we propose a novel nonlinear reconstruction method by adopting deep neural networks for representation. We then extend the method to a semi-supervised community detection algorithm by incorporating pairwise constraints among graph nodes. Extensive experimental results on synthetic and real networks show that the new methods are effective, outperforming most state-of-the-art methods for community detection. Community detection is an important technique to understand structures and patterns in complex networks. Recently, overlapping community detection becomes a trend due to the ubiquity of overlapping and nested communities in real world. However, existing approaches have ignored the use of implicit link preference information, i.e., links can reflect a node's preference on the targets of connections it wants to build. This information has strong impact on community detection since a node prefers to build links with nodes inside its community than those outside its community. In this paper, we propose a preference-based nonnegative matrix factorization (PNMF) model to incorporate implicit link preference information. Unlike conventional matrix factorization approaches, which simply approximate the original adjacency matrix in value, our model maximizes the likelihood of the preference order for each node by following the intuition that a node prefers its neighbors than other nodes. Our model overcomes the indiscriminate penalty problem in which non-linked pairs inside one community are equally penalized in objective functions as those across two communities. We propose a learning algorithm which can learn a node-community membership matrix via stochastic gradient descent with bootstrap sampling. We evaluate our PNMF model on several real-world networks. Experimental results show that our model outperforms state-of-the-art approaches and can be applied to large datasets. Community detection is a fundamental statistical problem in network data analysis. Many algorithms have been proposed to tackle this problem. Most of these algorithms are not guaranteed to achieve the statistical optimality of the problem, while procedures that achieve information theoretic limits for general parameter spaces are not computationally tractable. In this paper, we present a computationally feasible two-stage method that achieves optimal statistical performance in misclassification proportion for stochastic block model under weak regularity conditions. Our two-stage procedure consists of a generic refinement step that can take a wide range of weakly consistent community detection procedures as initializer, to which the refinement stage applies and outputs a community assignment achieving optimal misclassification proportion with high probability. The practical effectiveness of the new algorithm is demonstrated by competitive numerical results. 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. Community detection in the presence of prior information or preferences on solution properties is called semi-supervised or constrained community detection. The task of embedding such existing kinds of knowledge effectively within a community discovery algorithm is challenging. Indeed existing approaches are not flexible enough to incorporate a variety of background information types. This paper provides a framework for semi-supervised community detection based on constraint programming modelling technology for simultaneously modelling different objective functions such as modularity and a comprehensive range of constraint types including community level, instance level, definition based and complex logic constraints. An advantage of the proposed framework is that, using appropriate solvers, optimality can be established for the solutions found. Experiments on real and benchmark data sets show strong performance and flexibility for our proposed framework. Recently community detection has attracted much interest in social media to understand the collective behaviours of users and allow individuals to be modeled in the context of the group. Most existing approaches for community detection exploit either users' social links or their published content, aiming at discovering groups of densely connected or highly similar users. They often fail to find effective communities due to excessive noise in content, sparsity in links, and heterogenous behaviours of users in social media. Further, they are unable to provide insights and rationales behind the formation of the group and the collective behaviours of the users. To tackle these challenges, we propose to discover communities in a low- dimensional latent space in which we simultaneously learn the representation of users and communities. In particular, we integrated different social views of the network into a low-dimensional latent space in which we sought dense clusters of users as communities. By imposing a Laplacian regularizer into affiliation matrix, we further incorporated prior knowledge into the community discovery process. Finally community profiles were computed by a linear operator integrating the profiles of members. Taking the wellness domain as an example, we conducted experiments on a large scale real world dataset of users tweeting about diabetes and its related concepts, which demonstrate the effectiveness of our approach in discovering and profiling user communities. In this paper we present an original approach for community detection in complex networks. The approach belongs to the family of seed-centric algorithms. However, instead of expanding communities around selected seeds as most of existing approaches do, we explore here applying an ensemble clustering approach to different network partitions derived from ego-centered communities computed for each selected seed. Ego-centered communities are themselves computed applying a recently proposed ensemble ranking based approach that allow to efficiently combine various local modularities used to guide a greedy optimization process. Results of first experiments on real world networks for which a ground truth decomposition into communities are known, argue for the validity of our approach. 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. 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. Community detection, which aims to cluster @math nodes in a given graph into @math distinct groups based on the observed undirected edges, is an important problem in network data analysis. In this paper, the popular stochastic block model (SBM) is extended to the generalized stochastic block model (GSBM) that allows for adversarial outlier nodes, which are connected with the other nodes in the graph in an arbitrary way. Under this model, we introduce a procedure using convex optimization followed by @math -means algorithm with @math . Both theoretical and numerical properties of the method are analyzed. A theoretical guarantee is given for the procedure to accurately detect the communities with small misclassification rate under the setting where the number of clusters can grow with @math . This theoretical result admits to the best-known result in the literature of computationally feasible community detection in SBM without outliers. Numerical results show that our method is both computationally fast and robust to different kinds of outliers, while some popular computationally fast community detection algorithms, such as spectral clustering applied to adjacency matrices or graph Laplacians, may fail to retrieve the major clusters due to a small portion of outliers. We apply a slight modification of our method to a political blogs data set, showing that our method is competent in practice and comparable to existing computationally feasible methods in the literature. To the best of the authors' knowledge, our result is the first in the literature in terms of clustering communities with fast growing numbers under the GSBM where a portion of arbitrary outlier nodes exist. Community detection is an important task in network analysis. A community (also referred to as a cluster) is a set of cohesive vertices that have more connections inside the set than outside. In many social and information networks, these communities naturally overlap. For instance, in a social network, each vertex in a graph corresponds to an individual who usually participates in multiple communities. In this paper, we propose an efficient overlapping community detection algorithm using a seed expansion approach. The key idea of our algorithm is to find good seeds, and then greedily expand these seeds based on a community metric. Within this seed expansion method, we investigate the problem of how to determine good seed nodes in a graph. In particular, we develop new seeding strategies for a personalized PageRank clustering scheme that optimizes the conductance community score. An important step in our method is the neighborhood inflation step where seeds are modified to represent their entire vertex neighborhood. Experimental results show that our seed expansion algorithm outperforms other state-of-the-art overlapping community detection methods in terms of producing cohesive clusters and identifying ground-truth communities. We also show that our new seeding strategies are better than existing strategies, and are thus effective in finding good overlapping communities in real-world networks.