On Endogenous Random Consensus and Averaging Dynamics

Motivated by various random variations of the Hegselmann-Krause model for opinion dynamics and gossip algorithm in an endogenously changing environment, we propose a general framework for the study of endogenously varying random averaging dynamics, that is, averaging dynamics whose evolution suffers from history-dependent sources of randomness. We show that under general assumptions, such dynamics is convergent almost surely. We also determine the limiting behavior and show that infinitely many time-varying Lyapunov functions are admitted.