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 𝑖 𝑗 that models the relationship between 𝑖, 𝑗. The system evolves as opinions and edge weights are updated over time by the following rules:

• Opinion dynamics: The opinion of agent 𝑖 is updated as a linear combination of its current opinion and the weighted sum of neighbors' opinions with coefficients in matrix 𝑊 = [𝑤 𝑖 𝑗 ].

• Appraisal dynamics: The appraisal 𝑤 𝑖 𝑗 is updated as a linear combination of its current value and the agreement of the opinions of agents 𝑖 and 𝑗. The agreement of opinion 𝑣 𝑖 and 𝑣 𝑗 is taken as the dot product 𝑣 𝑖 • 𝑣 𝑗 . 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), • • • , 𝑣 𝑛 (𝑡)] and social tie weights 𝑊 (𝑡) = [𝑤 𝑖 𝑗 (𝑡)] 𝑛×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 𝑖 𝑗 𝑤 𝑗𝑘 𝑤 𝑘𝑖 ≥ 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 →∞ |𝑉 (𝑡)| = 0.

We also performed extensive simulations to examine how different initial conditions affect the network evolution. Of particular interest is that our dynamic model can be used to faithfully detect community structures. On real-world graphs, with a small number of seeds initially assigned ground truth opinions, the dynamic model successfully discovers the final community structure. The model sheds lights on why community structure emerges and becomes a widely observed, sustainable property in complex networks.