Echoes of Disagreement: Measuring Disparity in Social Consensus

TL;DR

This work introduces a disparity measure that captures how two social groups differently shape a consensus under the DeGroot and Friedkin-Johnsen models. It develops provable algorithms for disparity minimization and maximization, obtaining poly-time solutions for most cases while proving NP-hardness for optimal partitioning in DeGroot and deriving spectral-structure results for FJ. The authors analyze how intrinsic opinions, graph topology, and sentiment balance influence disparity, including bounds tied to Laplacian eigenvalues and Fiedler values, and show how regulator interventions via link-strength adjustments can reduce disparity. Experiments on real-world networks validate the theoretical findings and illustrate how assortativity and spectral properties govern disparity dynamics, with practical implications for designing interventions to promote equitable consensus.

Abstract

Public discourse and opinions stem from multiple social groups. Each group has beliefs about a topic (such as vaccination, abortion, gay marriage, etc.), and opinions are exchanged and blended to produce consensus. A particular measure of interest corresponds to measuring the influence of each group on the consensus and the disparity between groups on the extent to which they influence the consensus. In this paper, we study and give provable algorithms for optimizing the disparity under the DeGroot or the Friedkin-Johnsen models of opinion dynamics. Our findings provide simple poly-time algorithms to optimize disparity for most cases, fully characterize the instances that optimize disparity, and show how simple interventions such as contracting vertices or adding links affect disparity. Finally, we test our developed algorithms in a variety of real-world datasets.

Figures (2)

• Figure 1: Disparity Minimizing Markov Chain weights for the Karate Club Network. The node labels in (a) correspond to the intrinsic opinions of the nodes where $s = s' / \| s' \|$ with $s' \sim \mathcal{U} \left ([0, 1]^n \right )$. The edge colors correspond to the values of $T_{ij}$. The node labels in (b) correspond to a polarized network where each node in partition $A$ has an intrinsic opinion $s_i \sim \mathrm{Beta}(2, 8)$ and each node in partition $B$ has an intrinsic opinion $s_i \sim \mathrm{Beta}(8, 2)$. The node colors correspond to how far each node is from the average opinion. We observe that the high-value weights in the non-polarized network are amplified in the polarized network, and the low-value weights are de-amplified. Moreover, the Markov Chain defined according to the polarized network mixes slower than the non-polarized one.
• Figure 2: Changes in the normalized edge weights for the FJ model assuming balanced opinions and partitions (Theorem \ref{['theorem:change_edge_weights']}) for $T^* = \mathop{\mathrm{arg\,min}}\limits_{T : T \ge \mathbf 0, T \mathbf 1 = \mathbf 1, \mathrm{supp}(T) = G} \mathbb E_{s, A} \left [ f^{\mathsf{FJ}} (s, A, T) \right ]$. The blue (resp. red) edges in the Figure correspond to edges whose weight increased (resp. decreased) in the original graph. The new weights have been found by applying gradient descent using the gradients found in Theorem \ref{['theorem:change_edge_weights']}.

Theorems & Definitions (11)

• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 4
• Theorem 5
• Theorem 6
• Theorem 7
• Lemma 1
• proof
• Theorem 8