Agreement dynamics on small-world networks

¹ Laboratoire de Physique Théorique (CNRS UMR8627), Bâtiment 210 Université Paris-Sud - 91405 Orsay cedex, France
² Dipartimento di Fisica, Università “La Sapienza” and SMC-INFM P.le A. Moro 2, 00185 Roma, Italy

Received: 1 December 2005
Accepted: 26 January 2006

Abstract

In this paper we analyze the effect of a non-trivial topology on the dynamics of the so-called Naming Game, a recently introduced model which addresses the issue of how shared conventions emerge spontaneously in a population of agents. We consider in particular the small-world topology and study the convergence towards the global agreement as a function of the population size N as well as of the parameter p which sets the rate of rewiring leading to the small-world network. As long as , there exists a crossover time scaling as which separates an early one-dimensional–like dynamics from a late-stage mean-field–like behavior. At the beginning of the process, the local quasi–one-dimensional topology induces a coarsening dynamics which allows for a minimization of the cognitive effort (memory) required to the agents. In the late stages, on the other hand, the mean-field–like topology leads to a speed-up of the convergence process with respect to the one-dimensional case.