Fair Community Detection and Structure Learning in Heterogeneous Graphical Models

TL;DR

This paper introduces a novel fair graphical model selection method that ensures demographic groups are fairly represented in communities, using regularized pseudo-likelihood and convex optimization, with proven statistical consistency.

Contribution

It proposes a new $ ext{ extonehalf}$-regularized approach for fair community detection in graphical models, applicable to both Gaussian and Ising models, with theoretical guarantees.

Findings

Method recovers fair communities with high probability

Proposed approach is statistically consistent

Applicable to both continuous and binary data

Abstract

Inference of community structure in probabilistic graphical models may not be consistent with fairness constraints when nodes have demographic attributes. Certain demographics may be over-represented in some detected communities and under-represented in others. This paper defines a novel $\ell_1$-regularized pseudo-likelihood approach for fair graphical model selection. In particular, we assume there is some community or clustering structure in the true underlying graph, and we seek to learn a sparse undirected graph and its communities from the data such that demographic groups are fairly represented within the communities. In the case when the graph is known a priori, we provide a convex semidefinite programming approach for fair community detection. We establish the statistical consistency of the proposed method for both a Gaussian graphical model and an Ising model for, respectively, continuous and binary data, proving that our method can recover the graphs and their fair communities with high probability.