  We study transient behavior of gossip opinion dynamics, in which agents
randomly interact pairwise over a weighted graph with two communities. Edges
within a community have identical weights different from edge weights between
communities. We first derive an upper bound for the second moment of agent
opinions. Using this bound, we obtain upper bounds for probability that a large
proportion of agents have opinions close to average ones. The results imply a
phase transition of transient behavior of the process: When edge weights within
communities are larger than those between communities and those between regular
and stubborn agents, most agents in the same community hold opinions close to
the average opinion of that community with large probability, at an early stage
of the process. However, if the difference between intra- and inter-community
weights is small, most of the agents instead hold opinions close to everyone's
average opinion at the early stage. In contrast, when the influence of stubborn
agents is large, agent opinions settle quickly to steady state. We then conduct
numerical experiments to validate the theoretical results. Different from
traditional asymptotic analysis in most opinion dynamics literature, the paper
characterizes the influences of stubborn agents and community structure on the
initial phase of the opinion evolution.
