On existence of a quadratic comparison function for random weighted averaging dynamics and its implications

In this paper we study the stability and limiting behavior of discrete-time deterministic and random weighted averaging dynamics. We show that any such dynamics admits infinitely many comparison functions including a quadratic one. Using a quadratic comparison function, we establish the stability and characterize the set of equilibrium points of a broad class of random and deterministic averaging dynamics. This class includes a set of balanced chains, which itself contains many of the previously studied chains. Finally, we provide some implications of the developed results for products of independent random stochastic matrices.