Three-state opinion formation model on adaptive networks and time to consensus

We investigate the three-state majority rule model in a coevolving network with intensive average degree using Monte Carlo simulations. The key parameter investigated is the degree of homophily (heterophily), which is the probability p ( $q = 1 - p$ ) of a given person being affected by others with the same (different) opinion. For a system with a uniformly random initial state, so that each person has an equal chance of selecting one of the three opinions, based on extensive Monte Carlo simulations, we found that there are three distinct phases: (1) When the population has an intermediate homophilic tendency, it reaches the consensus state very fast. (2) When the system has a moderate to large heterophilic tendency (a small value of p), the time to consensus (or convergence time) can be significantly longer. (3) When the system has a high homophilic tendency (a large value of p), the population can remain in a polarization state for a long time. We defined the convergence time for the system of voters to reach consensus operationally, and obtained a distribution function for the convergence time through Monte Carlo simulations. We observed that the average convergence time in a three-state opinion formation process is generally faster than in the same model of voting dynamics when only two states are available to voters, given that in the beginning of the simulations, different opinions are uniformly spread in the population. The implications in diversity are discussed