# Voter model on networks partitioned into two cliques of arbitrary sizes

## TL;DR

This paper analyzes how the structure of two interconnected cliques affects the time it takes for opinions to reach consensus, revealing that optimal interconnections scale as N^{3/2} to minimize consensus time.

## Contribution

It provides a novel analysis of the voter model on networks with two cliques connected by variable edges, identifying how interclique links influence consensus time.

## Key findings

- Consensus time is not monotonically decreasing with interclique edges.
- Optimal interclique connections scale as N^{3/2} for fastest consensus.
- Equation-based analysis matches simulation results across all interconnection levels.

## Abstract

The voter model is an archetypal stochastic process that represents opinion dynamics. In each update, one agent is chosen uniformly at random. The selected agent then copies the current opinion of a randomly selected neighbour. We investigate 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 a monotonically decreasing function of $X$. Cliques of fixed proportions with opposite 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 more members in the other clique as $N$ increases but not to the extent that cliques lose their identity as distinct communities. We support our numerical results with an equation-based analysis. By interpolating between two asymptotic heterogeneous mean-field approximations, we obtain an equation for the mean consensus time that is in excellent agreement with simulations for all values of $X$.