Finite Time Bounds for Stochastic Bounded Confidence Dynamics

Stochastic bounded confidence (SBC) dynamics is a recently proposed framework that incorporates linear graph-based dynamics and bounded confidence dynamics as special cases, capturing the inherent stochasticity and noise in real-life interactions. The asymptotic analysis of SBC dynamics in prior work is a significant analytical contribution. However, such analyses do not shed light on the finite-time behavior of the dynamics, which is often crucial, e.g., in marketing campaigns, election canvassing, etc. In this paper, we present the first finite-time analysis of the existing SBC dynamics. The finite-time analysis of SBC presents some technical challenges due mainly to the inherent nonlinearity and stochasticity. We first characterize the finite-time behavior of SBC dynamics of two agents and then extend our analysis to a bistar social graph representing bipartisan democracies and commercial duopolies. Moreover, finite-time analysis of SBC requires us to develop concentration bounds for asymptotically zero-drift unbounded jump Markov processes, which, to our knowledge, have not been studied before. We derive high-probability bounds for the opinion difference between any two agents and demonstrate a strong concentration of opinion difference around zero under the exact conditions that ensure asymptotic stability of the dynamics. We also provide numerical experiments to compare with our analytical results.