Detecting Communities in a Gossip Model with Stubborn Agents

We consider a community detection problem for a gossip model, in which agents randomly interact pairwise, and there are stubborn agents never changing their states. Such a model can illustrate how disagreement and opinion fluctuation arise in a social network. It is assumed that each agent is assigned with one of the two community labels, and the agents interact with probabilities depending on their labels. The considered problem is twofold: to infer the community labels of agents, and to 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 detection and estimation algorithm 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. We derive a sample-complexity result for successful community detection, and analyze convergence rate of the interaction estimator. Simulations are presented for illustration of the performance of the proposed algorithm