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The voter model is an archetypal stochastic process that represents opiniondynamics. In each update, one agent is chosen uniformly at random. The selectedagent then copies the current opinion of a randomly selected neighbour. Weinvestigate the voter model on a network with an exogenous community structure:two cliques (i.e. complete subgraphs) randomly linked by $X$ interclique edges.We show that, counterintuitively, the mean consensus time is typically not amonotonically decreasing function of $X$. Cliques of fixed proportions withopposite initial opinions reach a consensus, on average, most quickly if $X$scales as $N^{3/2}$, where $N$ is the number of agents in the network. Hence,to accelerate a consensus between cliques, agents should connect to moremembers in the other clique as $N$ increases but not to the extent that cliqueslose their identity as distinct communities. We support our numerical resultswith an equation-based analysis. By interpolating between two asymptoticheterogeneous mean-field approximations, we obtain an equation for the meanconsensus time that is in excellent agreement with simulations for all valuesof $X$.

Voter model on networks partitioned into two cliques of arbitrary sizes