  We consider a community detection problem for gossip dynamics with stubborn
agents in this paper. It is assumed that the communication probability matrix
for agent pairs has a block structure. More specifically, we assume that the
network can be divided into two communities, and the communication probability
of two agents depends on whether they are in the same community. Stability of
the model is investigated, and expectation of stationary distribution is
characterized, indicating under the block assumption, the stationary behaviors
of agents in the same community are similar. It is also shown that agents in
different communities display distinct behaviors if and only if state averages
of stubborn agents in different communities are not identical. A community
detection algorithm is then proposed to recover community structure and to
estimate communication probability parameters. It is verified that the
community detection part converges in finite time, and the parameter estimation
part converges almost surely. Simulations are given to illustrate algorithm
performance.
