Opinion Dynamics in Heterogeneous Networks: Convergence Conjectures and Theorems

TL;DR

This paper investigates opinion dynamics in heterogeneous networks with state-dependent topologies, providing new convergence conditions and classifying equilibria in models based on confidence and influence ranges.

Contribution

It introduces a novel influence range model, classifies agents by topology, and establishes convergence conditions for opinion dynamics with state-dependent interactions.

Findings

Convergence to steady states under fixed topology is proven.

A positive invariant set guarantees convergence once entered.

The influence range model extends bounded confidence models.

Abstract

Recently, significant attention has been dedicated to the models of opinion dynamics in which opinions are described by real numbers, and agents update their opinions synchronously by averaging their neighbors' opinions. The neighbors of each agent can be defined as either (1) those agents whose opinions are in its "confidence range," or (2) those agents whose "influence range" contain the agent's opinion. The former definition is employed in Hegselmann and Krause's bounded confidence model, and the latter is novel here. As the confidence and influence ranges are distinct for each agent, the heterogeneous state-dependent interconnection topology leads to a poorly-understood complex dynamic behavior. In both models, we classify the agents via their interconnection topology and, accordingly, compute the equilibria of the system. Then, we define a positive invariant set centered at each equilibrium opinion vector. We show that if a trajectory enters one such set, then it converges to a steady state with constant interconnection topology. This result gives us a novel sufficient condition for both models to establish convergence, and is consistent with our conjecture that all trajectories of the bounded confidence and influence models eventually converge to a steady state under fixed topology.