A State Transition Matrix-Based Approach to Separation of Cooperations and Antagonisms in Opinion Dynamics

TL;DR

This paper introduces a novel state transition matrix approach to analyze opinion dynamics in networks with both cooperative and antagonistic interactions, accommodating switching topologies without strict connectivity assumptions.

Contribution

It develops a general framework linking signed digraphs and conventional digraphs, extending structural balance theory to dynamic networks for convergence analysis.

Findings

Bipartite consensus occurs under structurally balanced switching signed digraphs.

The approach applies to networks with changing topologies and no connectivity constraints.

It bridges opinion dynamics under signed and unsigned digraphs.

Abstract

This paper is concerned with the dynamics evolution of opinions in the presence of both cooperations and antagonisms. The class of Laplacian flows is addressed through signed digraphs subject to switching topologies. Further, a state transition matrix-based approach is developed for the analysis of opinion dynamics, regardless of any assumptions on connectivity, structural balance or digon sign-symmetry of signed digraphs. It is shown that based on the separation of cooperations and antagonisms, a relationship can be bridged between opinion dynamics under signed digraphs and under conventional digraphs. This helps to solve convergence problems for opinion dynamics. In particular, bipartite consensus (or stability) emerges if and only if the associated switching signed digraph is simultaneously structurally balanced (or unbalanced), which generalizes the use of structural balance theory in opinion dynamics to the case study of changing network topologies.