  Social sampling is a novel randomized message passing protocol inspired by
social communication for opinion formation in social networks. In a typical
social sampling algorithm, each agent holds a sample from the empirical
distribution of social opinions at initial time, and it collaborates with other
agents in a distributed manner to estimate the initial empirical distribution
by randomly sampling a message from current distribution estimate. In this
paper, we focus on analyzing the theoretical properties of the distributed
social sampling algorithm over random networks. Firstly, we provide a framework
based on stochastic approximation to study the asymptotic properties of the
algorithm. Then, under mild conditions, we prove that the estimates of all
agents converge to a common random distribution, which is composed of the
initial empirical distribution and the accumulation of quantized error.
Besides, by tuning algorithm parameters, we prove the strong consistency,
namely, the distribution estimates of agents almost surely converge to the
initial empirical distribution. Furthermore, the asymptotic normality of
estimation error generated by distributed social sampling algorithm is
addressed. Finally, we provide a numerical simulation to validate the
theoretical results of this paper.
