Topology-induced coarsening in language games

We investigate how very large populations are able to reach a global consensus, out of local microscopic interaction rules, in the framework of a recently introduced class of models of semiotic dynamics, the so-called naming game. We compare in particular the convergence mechanism for interacting agents embedded in a low-dimensional lattice with respect to the mean-field case. We highlight that in low dimensions consensus is reached through a coarsening process that requires less cognitive effort of the agents, with respect to the mean-field case, but takes longer to complete. In one dimension, the dynamics of the boundaries is mapped onto a truncated Markov process from which we analytically computed the diffusion coefficient. More generally we show that the convergence process requires a memory per agent scaling as N and lasts a time N1+2/d in dimension d <= 4 (the upper critical dimension), while in mean field both memory and time scale as N-3/2, for a population of N agents. We present analytical and numerical evidence supporting this picture.