  In this era of fast and large-scale opinion formation, a mathematical
understanding of opinion evolution, a.k.a. opinion dynamics, acquires
importance. Linear graph-based dynamics and bounded confidence dynamics are the
two popular models for opinion dynamics in social networks. Stochastic bounded
confidence (SBC) opinion dynamics was proposed as a general framework that
incorporates both these dynamics as special cases and also captures the
inherent stochasticity and noise (errors) in real-life social exchanges.
Although SBC dynamics is quite general and realistic, its analysis is more
challenging. This is because SBC dynamics is nonlinear and stochastic, and
belongs to the class of Markov processes that have asymptotically zero drift
and unbounded jumps. The asymptotic behavior of SBC dynamics was characterized
in prior works. However, they do not shed light on its finite-time behavior,
which is often of interest in practice. We take a stride in this direction by
analyzing the finite-time behavior of a two-agent system and a bistar graph,
which are crucial to the understanding of general multi-agent dynamics. In
particular, we show that the opinion difference between the two agents is
well-concentrated around zero under the conditions that lead to asymptotic
stability of the SBC dynamics.
