Co-evolution of Opinion and Social Tie Dynamics Towards Structural Balance

TL;DR

This paper introduces a co-evolution model for opinions and social ties, analyzing long-term behaviors like structural balance and opinion convergence, with explicit solutions in special cases.

Contribution

It provides a theoretical framework for understanding how opinions and social relationships co-evolve towards balance or neutral states, including explicit solutions for specific initial conditions.

Findings

Edge weights tend to stabilize with positive or negative signs.

The system reaches structural balance or opinions decay to zero.

Explicit solutions are derived when initial opinions align with eigenvectors of initial weights.

Abstract

In this paper, we propose co-evolution models for both dynamics of opinions (people's view on a particular topic) and dynamics of social appraisals (the approval or disapproval towards each other). Opinion dynamics and dynamics of signed networks, respectively, have been extensively studied. We propose a co-evolution model, where each vertex $i$ in the network has a current opinion vector $v_i$ and each edge $(i,j)$ has a weight $w_{ij}$ that models the relationship between $i,j$. The system evolves as the opinions and edge weights are updated over time by the following rules, Opinion Dynamics and Appraisal Dynamics. We are interested in characterizing the long-time behavior of the dynamic model -- i.e., whether edge weights evolve to have stable signs (positive or negative) and structural balance (the multiplication of weights on any triangle is non-negative). Our main theoretical result solves the above dynamic system with time-evolving opinions $V(t)=[v1(t),\cdots,vn(t)]$ and social tie weights $W(t)=[w_{ij}(t)]_{n\times n}$. For a generic initial opinion vector $V(0)$ and weight matrix $W(0)$, one of the two phenomena must occur at the limit. The first one is that both sign stability and structural balance (for any triangle with individual $i,j,k$, $w_{ij}w_{jk}w_{ki}\geq0$) occur. In the special case that $V(0)$ is an eigenvector of $W(0)$, we are able to obtain the explicit solution to the co-evolution equation and give exact estimates on the blowup time and rate convergence. The second one is that all the opinions converge to $0$, i.e., $\lim_{t\rightarrow \infty}|V(t)|=0$.