The Geometry of Community Detection via the MMSE Matrix

Abstract

The information-theoretic limits of community detection have been studied extensively for network models with high levels of symmetry or homogeneity. The contribution of this paper is to study a broader class of network models that allow for variability in the sizes and behaviors of the different communities, and thus better reflect the behaviors observed in real-world networks. Our results show that the ability to detect communities can be described succinctly in terms of a matrix of effective signal-to-noise ratios that provides a geometrical representation of the relationships between the different communities. This characterization follows from a matrix version of the I-MMSE relationship and generalizes the concept of an effective scalar signal-to-noise ratio introduced in previous work. We provide explicit formulas for the asymptotic per-node mutual information and upper bounds on the minimum mean-squared error. The theoretical results are supported by numerical simulations. * G. Reeves is with the

Mutual Information in Community Detection with Covariate Information and Correlated Networks

“…Without loss of generality these labels can be embedded into finite dimensional Euclidean space. To facilitate the exposition of our results, we use the whitened representation described in [13], where the labels are supported on a set of k points in {µ 1 , . .…”

“…A unique specification of this whitened representation is described in [13,Remark 1]. There are two types of the covariate information.…”

“…Each network is represented by an n × n binary adjacency matrix G ℓ = (G ℓ ij ) where G ℓ ij = G ℓ ji = 1 if there is an edge between nodes i and j and zero otherwise. Following [13], each network is drawn according to a degree-balanced SBM of the form…”

“…The information-theoretic limits describe the optimal performance that can be attained without any constraints on computational complexity. For network models with k ≥ 4 communities [2,5,20] or asymmetries [5,13], there exists a computational-to-statistical gap, where weak recovery below the Kesten-Stigum threshold is information-theoretically possible, even though all known polynomial-time algorithms fail in this regime.…”

“…General SBMs have been considered by Abbe and Sandon [10] who characterize conditions for weak recovery, Lesieur et al [7] who analyze the performance of an approximate message passing algorithm, and Reeves et. al [13] who study the asymptotic per-node mutual information and MMSE in degree-balanced SBMs.…”