This paper presents a novel spectral algorithm with additive clustering designed to identify overlapping communities in networks. The algorithm is based on geometric properties of the spectrum of the expected adjacency matrix in a random graph model that we call stochastic blockmodel with overlap (SBMO). An adaptive version of the algorithm, that does not require the knowledge of the number of hidden communities, is proved to be consistent under the SBMO when the degrees in the graph are (slightly more than) logarithmic. The algorithm is shown to perform well on simulated data and on real-world graphs with known overlapping communities.
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. Identification of (overlapping) communities clusters in a complex network is a general problem in data mining of network data sets. In this paper, we devise a novel algorithm to identify overlapping communities in complex networks by the combination of a new modularity function based on generalizing NG's Q function, an approximation mapping of network nodes into Euclidean space and fuzzy c-means clustering. Experimental results indicate that the new algorithm is efficient at detecting both good clusterings and the appropriate number of clusters. 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. Community detection is a fundamental problem in network analysis, with applications in many diverse areas. The stochastic block model is a common tool for model-based community detection, and asymptotic tools for checking consistency of community detection under the block model have been recently developed. However, the block model is limited by its assumption that all nodes within a community are stochastically equivalent, and provides a poor fit to networks with hubs or highly varying node degrees within communities, which are common in practice. The degree-corrected stochastic block model was proposed to address this shortcoming and allows variation in node degrees within a community while preserving the overall block community structure. In this paper we establish general theory for checking consistency of community detection under the degree-corrected stochastic block model and compare several community detection criteria under both the standard and the degree-corrected models. We show which criteria are consistent under which models and constraints, as well as compare their relative performance in practice. We find that methods based on the degree-corrected block model, which includes the standard block model as a special case, are consistent under a wider class of models and that modularity-type methods require parameter constraints for consistency, whereas likelihood-based methods do not. On the other hand, in practice, the degree correction involves estimating many more parameters, and empirically we find it is only worth doing if the node degrees within communities are indeed highly variable. We illustrate the methods on simulated networks and on a network of political blogs.