Convergence Analysis of Weighted Median Opinion Dynamics with Higher-Order Effects

TL;DR

The paper addresses how to model public opinion formation when external environmental factors exert higher-order, group-level influences. It introduces a discrete-time, synchronous-weighted-median opinion dynamics model on a simplicial complex, integrating direct neighbor interactions with indirect environmental effects via environment-weighted medians and a row-stochastic indicator matrix A. The authors provide rigorous convergence results: (i) in heterogeneous populations with both opinionated and unopinionated agents, they derive a sufficient condition for asymptotic consensus; (ii) in fully opinionated populations, they prove convergence and exponential rates via a contraction mapping, along with a matrix-based expression for the limit point. Simulations on heterogeneous and homogeneous systems corroborate the theory and demonstrate how adjusting high-order weights can enforce or disrupt consensus, highlighting the practical impact of environment-driven higher-order interactions in opinion dynamics.

Abstract

The weighted median mechanism provides a robust alternative to weighted averaging in opinion dynamics. Existing models, however, are predominantly formulated on pairwise interaction graphs, which limits their ability to represent higher-order environmental effects. In this work, a generalized weighted median opinion dynamics model is proposed by incorporating high-order interactions through a simplicial complex representation. The resulting dynamics are formulated as a nonlinear discrete-time system with synchronous opinion updates, in which intrinsic agent interactions and external environmental influences are jointly modeled. Sufficient conditions for asymptotic consensus are established for heterogeneous systems composed of opinionated and unopinionated agents. For homogeneous opinionated systems, convergence and convergence rates are rigorously analyzed using the Banach fixed-point theorem. Theoretical results demonstrate the stability of the proposed dynamics under mild conditions, and numerical simulations are provided to corroborate the analysis. This work extends median-based opinion dynamics to high-order interaction settings and provides a system-level framework for stability and consensus analysis.

Figures (6)

• Figure 1: The visualization example presents the simplicial complex considered in the simulation part, which includes $10$ agents and $3$ simplices: $C_{1}$ is a $3$-simplex, $C_{2}$ and $C_{3}$ are $2$-simplex, where $\delta_{C_{1}}=\{7, 8, 9, 10\}$, $\delta_{C_{2}}=\{1, 2, 3\}$ and $\delta_{C_{3}}=\{4, 5, 6\}$. The existence of an arrow between two nodes in the graph indicates that one agent will affect the other agent, and the number near the arrow represents the influence agent weight. In addition, this simplicial complex ignores the internal connections within the $C_{1}$ simplex for convenience of drawing.
• Figure 2: Simplicial complexes of the simulation example of different system. There are three simplexes in the system, which are $\delta_{C_{1}}=\{7, 8, 9, 10\}$, $\delta_{C_{2}}=\{1, 2, 3\}$ and $\delta_{C_{3}}=\{4, 5, 6\}$. (a) Heterogeneous system: $\delta_{C_{1}}$ is a simplex composed of unopinionated agents, while $\delta_{C_{2}}$ and $\delta_{C_{3}}$ are simplices composed of opinionated agents. Furthermore, according to the weight of agents, $C_{1}$ is a cohesive agent set composed of unopinionated agents. (b) Homogeneous system: $\delta_{C_{1}}$, $\delta_{C_{2}}$ and $\delta_{C_{3}}$ are simplices composed of opinionated agent. In both systems, the following notations and rules apply consistently: Black arrows represent the influence between agents, and colored arrows represent the influence of simplices on agent. The numbers near the arrows represent the influence weights. If no number is marked, the influence weight is 1. Nodes in the red-covered area represent agents with bias, while those in the green-covered area represent unopinionated agents.
• Figure 3: Illustration of influence weights in different dimensions. To make Fig.\ref{['fig_a']}(a) meet the conditions of Theorem \ref{['T3.1']}, the low-order and high-order influence weights are modified. (a) Influence weight between agents: By adjusting the weights between agent 7 and its neighbors, where dashed arrows represent the adjusted weights, the cohesive agent set $C_{1}$ composed of unopinionated agents in Fig.\ref{['fig_a']}(a) is disrupted. (b) Influence weight of the environment on agents: By changing the influence weights of the simplex on agents, the system forms a weak cohesive group set $\{C_{2}, C_{3}\}$ composed of opinionated agents.
• Figure 4: The evolution processes of opinion for heterogeneous system, starting from different initial opinions of agents at time 0. Red represents opinionated agents with the same bias, green represents unopinionated agents, and the dashed line represents the common bias of all opinionated agents. (a) Before weight adjustment, the system cannot reach a consensus, but instead forms two stable states. (b) After the weight adjustment, the system can reach a consensus, which does not contain a cohesive agent set composed of unopinionated agents and contain a weak cohesive group set composed of opinionated agents.
• Figure 5: The evolution processes of opinion for homogeneous system, starting from diverse initial opinions of agents at time 0. All agents in this system are opinionated, with different colors representing distinct opinionated agents. (a) Each agent possesses a different bias, and dashed lines represent the bias values of agents. (b) Each agent shares the same bias value.

Theorems & Definitions (22)

• Definition 2.1
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Definition 3.4
• Definition 3.5
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4