  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.
