Modularity, asymmetry, and polarization shape consensus speed in the voter model

Abstract

In populations with community structure, the formation of consensus requires both alignment within and diffusion of beliefs across groups, processes that evolve on distinct time scales. 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 of sizes $N_1$ and $N_2$. At each time step, a pair of nodes is selected; if their binary opinions differ, each agent adopts the opinion of the other with probability $p$. With probability $\alpha$, the pairing occurs with a single clique, and with probability $1-\alpha$, across cliques. We analyze how this coupling strength, population imbalance, and initial polarization jointly determine the time to consensus. Formation of consensus generally starts with inter-clique interactions rapidly synchronizing the two cliques' opinion fractions, after which consensus is reached through a slower diffusion along the synchronized manifold; this slow stage is largely insensitive to $\alpha$ except when the cliques are nearly disconnected. To analyze these dynamics, we derive stochastic differential equations and Fokker-Planck approximations in the large-population limit, and assess their accuracy against the discrete model. While $\alpha$ primarily affects the fast alignment 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, and provides an approximate decomposition of consensus time into fast and slow contributions.