Local and global ordering dynamics in multistate voter models

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.