Consistency of maximum-likelihood and variational estimators in the stochastic block model

Abstract

AMS 2000 subject classifications: Primary 62G05, 62G20; secondary 62E17, 62H30.The stochastic block model (SBM) is a probabilistic model designed to describe heterogeneous directed and undirected graphs. In this paper, we address the asymptotic inference in SBM by use of maximum-likelihood and variational approaches. The identifiability of SBM is proved while asymptotic properties of maximum-likelihood and variational estimators are derived. In particular, the consistency of these estimators is settled for the probability of an edge between two vertices (and for the group proportions at the price of an additional assumption), which is to the best of our knowledge the first result of this type for variational estimators in random graphs

The $\beta$-Model—Maximum Likelihood, Cramér–Rao Bounds, and Hypothesis Testing

“…The FIM in (21) can now be obtained from (35). Under the assumptions preceding (22), it follows that…”

“…Now, applying the special case of the Sherman-Morrison formula in (37) to the FIM in (36) while assuming n > 2 immediately gives the inverse FIM in (22). APPENDIX B Here, we derive the FIM in (24), given for the directed β-model.…”

“…For example, in a social network, a nodal covariate could represent gender or status, whereas a dyadic covariate could represent absolute age difference or spatial distance. Random graphs with nodal covariates are commonly modeled as stochastic block models (SBMs) [19]- [22]. In a SBM, the nodes are divided into groups (blocks), with each group collecting all nodes with some given covariate values.…”

A scalable community detection algorithm for large graphs using stochastic block models

“…We also note that a node will take only a block label from its neighbors, and the time to enumerate over different choices of r becomes O(M i ) for node i in Eq. (19). Thus, the time complexity of computing labels for all the node is reduced to …”

“…Stochastic block models (SBMs) are among the important probabilistic tools describing the connectivity relationship between pairs of nodes [18], and have received considerable attention both in theoretical [19] and application domains [20].…”