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

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 $V(t)=[v_1(t), \cdots, v_n(t)]$ 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 $i, j, k$, $w_{ij}w_{jk}w_{ki}\geq 0$) 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., $\lim_{t\rightarrow \infty}|V(t)|=0$.