In populations with community structure, the formation of consensus requires both alignment within and diffusion of beliefs across groups, processes that evolve on distinct timescales. How do modularity, asymmetry, and polarization shape this process? We study a variant of the voter model in which a population is divided into two cliques. At each time step, a pair of nodes is selected; if their binary opinions differ, then each agent adopts the opinion of the other with a certain probability. The type of pair selected depends on a network structure parameter—modularity that is central in this paper. We analyze how this coupling strength, population imbalance, and initial polarization jointly determine the time to consensus. Formation of consensus generally starts with interclique interactions rapidly synchronizing the two cliques' opinion fractions, after which consensus is reached through a slower diffusion along the synchronized curve. This slow stage is largely insensitive to modularity except when the cliques are nearly disconnected. To analyze these dynamics, we derive time-continuous approximations in the large-population limit and assess their accuracy against the discrete model. While modularity primarily affects the synchronization stage, initially polarized and asymmetric populations exhibit nontrivial effects, including regimes in which an intermediate level coupling minimizes consensus time. A small-clique scaling analysis reveals that this optimum arises from a competition between fast alignment drift and noise amplification in the smaller group.

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• Consensus & non-consensus models
• Fokker–Planck equation
• Network Models
• Stochastic differential equations
• Voter model