Finding planted partitions in random graphs with general degree distributions

We consider the problem of recovering a planted partition such as a coloring, a small bisection, or a large cut in an (apart from that) random graph. In the last 30 years many algorithms for this problem have been developed that work provably well on various random graph models resembling the Erdős–Rényi model Gn,m. In these random graph models edges are distributed uniformly, and thus the degree distribution is very regular. By contrast, the recent theory of large networks shows that real-world networks frequently have a significantly different distribution of the edges and hence also a different degree distribution. Therefore, a variety of new types of random graphs have been introduced to capture these specific properties. One of the most popular models is characterized by a prescribed expected degree sequence. We study a natural variant of this model that features a planted partition. Our main result is that there is a polynomial time algorithm for recovering (a large part of) the planted partition in this model even in the sparse case, where the average degree is constant. In contrast to prior work, the input of the algorithm consists only of the graph, i.e., no further parameters of the model (such as the expected degree sequence) are revealed to the algorithm

A spectral method for community detection in moderately-sparse degree-corrected stochastic block models

We consider community detection in Degree-Corrected Stochastic Block Models (DC-SBM). We propose a spectral clustering algorithm based on a suitably normalized adjacency matrix. We show that this algorithm consistently recovers the block-membership of all but a vanishing fraction of nodes, in the regime where the lowest degree is of order log$(n)$ or higher. Recovery succeeds even for very heterogeneous degree-distributions. The used algorithm does not rely on parameters as input. In particular, it does not need to know the number of communities

Consistency of community detection in networks under degree-corrected stochastic block models

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.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1036 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org). With Correction