  We study how to jointly recover the community structure and estimate the
interaction probabilities of gossip opinion dynamics. In this process, agents
randomly interact pairwise, and there are stubborn agents never changing their
states. Such a model illustrates how disagreement and opinion fluctuation arise
in a social network. It is assumed that each agent is assigned with one of two
community labels, and the agents interact with probabilities depending on their
labels. The considered problem is to jointly recover the community labels of
the agents and estimate interaction probabilities between the agents, based on
a single trajectory of the model. We first study stability and limit theorems
of the model, and then propose a joint recovery and estimation algorithm based
on a trajectory. It is verified that the community recovery can be achieved in
finite time, and the interaction estimator converges almost surely. We derive a
sample-complexity result for the recovery, and analyze the estimator's
convergence rate. Simulations are presented for illustration of the performance
of the proposed algorithm.
