The motivation for this paper is to apply Bayesian structure learning using Model Averaging in large-scale networks. Currently, Bayesian model averaging algorithm is applicable to networks with only tens of variables, restrained by its super-exponential complexity. We present a novel framework, called LSBN(Large-Scale Bayesian Network), making it possible to handle networks with infinite size by following the principle of divide-and-conquer. The method of LSBN comprises three steps. In general, LSBN first performs the partition by using a second-order partition strategy, which achieves more robust results. LSBN conducts sampling and structure learning within each overlapping community after the community is isolated from other variables by Markov Blanket. Finally LSBN employs an efficient algorithm, to merge structures of overlapping communities into a whole. In comparison with other four state-of-art large-scale network structure learning algorithms such as ARACNE, PC, Greedy Search and MMHC, LSBN shows comparable results in five common benchmark datasets, evaluated by precision, recall and f-score. What's more, LSBN makes it possible to learn large-scale Bayesian structure by Model Averaging which used to be intractable. In summary, LSBN provides an scalable and parallel framework for the reconstruction of network structures. Besides, the complete information of overlapping communities serves as the byproduct, which could be used to mine meaningful clusters in biological networks, such as protein-protein-interaction network or gene regulatory network, as well as in social network.
One of traditional methods regards overlapping communities detection as an optimization problem, specifically, each community is identified as a subgraph reaching local optimization given quality function @math , thus detecting overlapping communities becomes finding all locally-optimized subgraphs @cite_23 . Furthermore, the optimization could be augmented by combination with spectral mapping and fuzzy clustering @cite_9 .
@cite_9 @cite_23
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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. 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. Given a large network, local community detection aims at finding the community that contains a set of query nodes and also maximizes (minimizes) a goodness metric. This problem has recently drawn intense research interest. Various goodness metrics have been proposed. However, most existing metrics tend to include irrelevant subgraphs in the detected local community. We refer to such irrelevant subgraphs as free riders. We systematically study the existing goodness metrics and provide theoretical explanations on why they may cause the free rider effect. We further develop a query biased node weighting scheme to reduce the free rider effect. In particular, each node is weighted by its proximity to the query node. We define a query biased density metric to integrate the edge and node weights. The query biased densest subgraph, which has the largest query biased density, will shift to the neighborhood of the query nodes after node weighting. We then formulate the query biased densest connected subgraph (QDC) problem, study its complexity, and provide efficient algorithms to solve it. We perform extensive experiments on a variety of real and synthetic networks to evaluate the effectiveness and efficiency of the proposed methods. 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.