Local and global ordering dynamics in multistate voter models

Ramirez, Lucía; San Miguel, Maxi; Galla, Tobias
PHYSICAL REVIEW E 106, 054307 (1-17) (2022)

We investigate the time evolution of the density of active links and of the entropy of the distribution of agents
among opinions in multistate voter models with all-to-all interaction and on uncorrelated networks. Individual
realizations undergo a sequence of eliminations of opinions until consensus is reached. After each elimination the
population remains in a metastable state. The density of active links and the entropy in these states varies from
realization to realization. Making some simple assumptions we are able to analytically calculate the average
density of active links and the average entropy in each of these states. We also show that, averaged over
realizations, the density of active links decays exponentially, with a timescale set by the size and geometry of the
graph, but independent of the initial number of opinion states. The decay of the average entropy is exponential
only at long times when there are at most two opinions left in the population. Finally, we show how metastable
states comprising only a subset of opinions can be artificially engineered by introducing precisely one zealot in
each of the prevailing opinions.

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