Fair Community Detection and Structure Learning in Heterogeneous Graphical Models

TL;DR

This paper defines a novel $\ell_1$-regularized pseudo-likelihood approach for fair graphical model selection, and assumes there is some community or clustering structure in the true underlying graph, and seeks to learn a sparse undirected graph and its communities from the data such that demographic groups are fairly represented within the communities.

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.

Figures (10)

• Figure 1: Examples of graphical models with $p = 10$ nodes and demographic groups $\mathcal{V}=\mathcal{D}_1\cup\mathcal{D}_2$, where $\mathcal{D}_1$ is the set of red nodes and $\mathcal{D}_2$ is the set of cyan nodes. The left and right figures show the graphical model renderings of the sparse precision matrices associated with ${\bf{Q}}'$ and ${\bf{Q}}$ (fair membership matrix) defined in Example \ref{['exam:fair:sbm']}, respectively. In the fair graphical model (right) with disjoint communities $\mathcal{C}_1$, $\mathcal{C}_2$, and $\mathcal{C}_3$, we have $|\mathcal{D}_h\cap\mathcal{C}_k|/|\mathcal{C}_k| = 1/2 = |\mathcal{D}_h|/p$ for all $k\in\{1,2,3\}$ and $h\in\{1,2\}$, as a result, ${\bf{Q}}$ satisfies demographic parity in every community—equivalently, ${\bf{R}}({\bf{I}}-{\bf{J}}_p/p){\bf{Q}}={\bf{0}}$. The right panel (membership with ${\bf{Q}}$) therefore represents a fairer model than the left panel (membership with ${\bf{Q}}'$), since ${\bf{Q}}'$ violates this balance in at least one community, i.e., ${\bf{R}}({\bf{I}}-{\bf{J}}_p/p){\bf{Q}}'\neq {\bf{0}}$. The dotted edges represent lower-probability connections across communities ($a=0.1$), while solid edges are higher-probability within-community connections ($5a$).
• Figure 2: Comparison of (a) our joint estimation framework with (b) a traditional two-step approach.
• Figure 3: CE and runtime (in seconds) of CD-I amini2018semidefinite, CD-II cai2015robust, and the proposed FCD on a stochastic block model (SBM) with $H = 5$ and $K \in \{5, 10\}$.
• Figure 4: Balance and RCut of CD-I, CD-II cai2015robust, FCD, GM-B.FI, and FCONCORD on Friendship (left) and DrugNet (right) Networks as a function of the number $k$ of clusters.
• Figure 5: Runtime (in seconds) of CD-I, CD-II cai2015robust, FCD, GM-B.FI, and FCONCORD on Friendship (left) and DrugNet (right) Networks as a function of the number $k$ of clusters.

Theorems & Definitions (32)

• Remark 1
• Theorem 2
• Remark 3
• Lemma 4
• Remark 5
• Theorem 6
• Theorem 7
• Theorem 8
• Example 9
• Lemma 10