We consider a community detection problem in a gossip model, where agents randomly interact pairwise, with stubborn agents never changing their states. It is assumed that the agents can be divided into two communities based on their interaction probability with others. Such a model can illustrate how disagreement and opinion fluctuation arise in a social network. The considered problem is twofold: to infer which community each agent belongs to, and to estimate interaction probabilities between agents, by only observing their state evolution. First, stability and limit theorems of the model are derived. An integrated detection and estimation algorithm is then proposed to infer the two communities and to estimate the interaction probabilities, based on agent states. It is verified that the community detector of the algorithm converges in finite time, and the interaction estimator converges almost surely. In addition, non-asymptotic property is obtained for the former, and convergence rate is analyzed for the latter. Simulations are presented to illustrate the performance of the proposed algorithm.
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