Optimal Activation in Asynchronous Opinion Dynamics

We study randomized asynchronous opinion dynamics in the Friedkin-Johnsen model, where agents update according to random activation. We establish a direct convergence proof with an explicit pathwise convergence rate and analyze its dependence on the activation probability vector. We then consider the problem of maximizing this rate. Although convexity of the objective remains open, the problem admits a reformulation with linear-bilinear constraints, which can be handled efficiently via sequential linear programming. In the single-agent update setting, the optimal activation distribution induces a novel centrality measure quantifying each agent’s importance for rapid equilibration. Numerical experiments show that this measure differs from classical centralities, while remaining correlated with influence-based ones.