Noisy Naming Games, partial synchronization and coarse-graining in social networks

The Naming Games (NG) are typical agent-based models for agreement dynamics, peer pressure and herding in social networks, and protocol selection in autonomous ad-hoc sensor networks. They form a large class that includes the Voter models and many others. By introducing a rare Poisson noise term to the signaling protocol of the NG, the resulting Markov Chain model called Noisy Naming Game (NNG) is ergodic, in which all partial consensus states are recurrent. By this generic method, any member of a large class of agent-based network models, including the Voter models and Bass models, is easily changed from a totally synchronous system to a partially synchronous one where even after reaching total consensus, the network revisits multi-namestates infinitely often. The method introduced here works on any underlying network topology including small world and scale-free ones. Furthermore, by organizing the partially-synchronized namestates / microstates according to their coarse-grained total probability, which counts the number of namestates associated with a macrostate / community structure (CS) weighted by their microstate probability, the NNG offers a new method for ranking competing CS in social interactions, that is not based entirely on hierarchical modularity or other ways for counting the number of intra vs inter group links in the social network. In fact, the coarse-grained CS incorporates entropic effects since its entropy is enumerated as the logarithm of the relative number of namestates it contains. As such, the Gibbs free energy is taken here to represent a good measure of overall social tension, arising from the ways in which different possibly overlapping subgroups choose and maintain differing opinions. Through simulations the NNG is shown to also successfully resolve the smallest groups.