Entropy of stochastic blockmodel ensembles

TL;DR

This paper derives entropy formulas for various stochastic blockmodel ensembles, including degree-corrected versions, and demonstrates their use in inferring latent network structures from observed data.

Contribution

It provides new entropy expressions for degree-corrected stochastic blockmodels and their variants, enabling improved inference of network block structures.

Findings

Derived entropy formulas for multiple ensemble variants

Demonstrated inference of latent block structure using entropy-based likelihood

Applicable to networks with intrinsic and extrinsic degree correlations

Abstract

Stochastic blockmodels are generative network models where the vertices are separated into discrete groups, and the probability of an edge existing between two vertices is determined solely by their group membership. In this paper, we derive expressions for the entropy of stochastic blockmodel ensembles. We consider several ensemble variants, including the traditional model as well as the newly introduced degree-corrected version [Karrer et al. Phys. Rev. E 83, 016107 (2011)], which imposes a degree sequence on the vertices, in addition to the block structure. The imposed degree sequence is implemented both as "soft" constraints, where only the expected degrees are imposed, and as "hard" constraints, where they are required to be the same on all samples of the ensemble. We also consider generalizations to multigraphs and directed graphs. We illustrate one of many applications of this measure by directly deriving a log-likelihood function from the entropy expression, and using it to infer latent block structure in observed data. Due to the general nature of the ensembles considered, the method works well for ensembles with intrinsic degree correlations (i.e. with entropic origin) as well as extrinsic degree correlations, which go beyond the block structure.