The discovery of community structures in social networks has gained significant attention since it is a fundamental problem in understanding the networks' topology and functions. However, most social network data are collected from partially observable networks with both missing nodes and edges. In this paper, we address a new problem of detecting overlapping community structures in the context of such an incomplete network, where communities in the network are allowed to overlap since nodes belong to multiple communities at once. To solve this problem, we introduce KroMFac, a new framework that conducts community detection via regularized nonnegative matrix factorization (NMF) based on the Kronecker graph model. Specifically, from a generative parameter matrix acquired by the expectation-maximization (EM) algorithm, we first estimate the missing part of the network. As our major contribution to the proposed framework, to improve community detection accuracy, we then characterize and select influential nodes (which tend to have high degrees) by ranking, and add them to the existing graph. Finally, we uncover the community structures by solving the regularized NMF-aided optimization problem in terms of maximizing the likelihood of the underlying graph. Furthermore, adopting normalized mutual information (NMI), we empirically show superiority of our KroMFac approach over two baseline schemes.
Community detection in incomplete networks. Recently, research on community detection in incomplete networks with missing edges has attracted wide attention due to a lack of information caused by users' privacy settings and limited resources. Most of these studies predict the missing links between nodes based on the incorporated additional information @cite_41 or the similarity of topological structures @cite_3 @cite_35 , and then discover communities in the underlying recovered networks. @cite_29 , a hierarchical gamma process infinite edge partition model was presented to detect communities and recover missing edges in parallel. In contrast to edge recovery, approaches for node recovery are largely still missing.
@cite_41 @cite_35 @cite_29 @cite_3
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With the recent advances in information networks, the problem of community detection has attracted much attention in the last decade. While network community detection has been ubiquitous, the task of collecting complete network data remains challenging in many real-world applications. Usually the collected network is incomplete with most of the edges missing. Commonly, in such networks, all nodes with attributes are available while only the edges within a few local regions of the network can be observed. In this paper, we study the problem of detecting communities in incomplete information networks with missing edges. We first learn a distance metric to reproduce the link-based distance between nodes from the observed edges in the local information regions. We then use the learned distance metric to estimate the distance between any pair of nodes in the network. A hierarchical clustering approach is proposed to detect communities within the incomplete information networks. Empirical studies on real-world information networks demonstrate that our proposed method can effectively detect community structures within incomplete information 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 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. Consider a network consisting of two subnetworks (communities) connected by some external edges. Given the network topology, the community detection problem can be cast as a graph partitioning problem that aims to identify the external edges as the graph cut that separates these two subnetworks. In this paper, we consider a general model where two arbitrarily connected subnetworks are connected by random external edges. Using random matrix theory and concentration inequalities, we show that when one performs community detection via spectral clustering there exists an abrupt phase transition as a function of the random external edge connection probability. Specifically, the community detection performance transitions from almost perfect detectability to low detectability near some critical value of the random external edge connection probability. We derive upper and lower bounds on the critical value and show that the bounds are equal to each other when two subnetwork sizes are identical. Using simulated and experimental data we show how these bounds can be empirically estimated to validate the detection reliability of any discovered communities.