External Bias and Opinion Clustering in Cooperative Networks

TL;DR

This work studies opinion dynamics on networks subject to unavoidable external biases and shows how a Laplacian-based model with a constant bias $\mathbf{b}$ and a controllable input $\mathbf{u}$ can realize prescribed clustering. It provides stability conditions for the biased system, derives explicit steady-state behavior for common graph classes, and presents a constructive control design to achieve any target opinion configuration. A key contribution is a two-stage and time-switching control strategy that expands the reachable set from the invariant subspace defined by the zero eigenvalues to the entire space $\mathbb{R}^n$, applicable to arbitrary graph topology. The results offer a principled way to counteract polarisation induced by biases and enable targeted opinion patterns in complex networks, with potential applications to task allocation, information campaigns, and social planning.

Abstract

In this work, we consider a group of n agents which interact with each other in a cooperative framework. A Laplacian-based model is proposed to govern the evolution of opinions in the group when the agents are subjected to external biases like agents' traits, news, etc. The objective of the paper is to design a control input which leads to any desired opinion clustering even in the presence of external bias factors. Further, we also determine the conditions which ensure the reachability to any arbitrary opinion states. Note that all of these results hold for any kind of graph structure. Finally, some numerical simulations are discussed to validate these results.

Figures (3)

• Figure 1: Graph topology
• Figure 2: Polarisation of opinion states
• Figure 3: Clustering of opinion states

Theorems & Definitions (5)

• Theorem 1
• Corollary 1
• Corollary 2
• Remark 1
• Theorem 2