  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.
