What is the Expected Transient Behavior of Opinion Evolution for Two Communities?

We study the transient behavior of a gossip model, in which agents randomly interact pairwise over a weighted graph with two communities. Edges within each community have identical weights, different from the weights between communities. It is shown that, at the early stage of the opinion evolution, the expected agent states in the same community have identical sign, despite influence of stubborn agents. Moreover, it is shown that the expected states of the agents in the same community concentrate around the initial average opinion of that community, if the weights within communities are larger than between. In contrast, if the edge weights between communities are larger, then the expected states of all agents concentrate around everyone's initial average opinion. Different from the traditional asymptotic analysis in the opinion dynamics literature, these results focus on the initial phase of opinion evolution and establish a correspondence between community structure and transient behavior of the gossip model. The results are illustrated by numerical examples.