On convergence and threshold properties of discrete Lotka-Volterra population protocols

Abstract

We study population protocols whose dynamics are modeled by the discrete Lotka-Volterra equations. Such protocols capture the dynamics of some opinion spreading models and generalize the Rock-Paper-Scissors discrete dynamics. Pairwise interactions among agents are scheduled uniformly at random. We consider convergence time and show that any such protocol on an n-agent population converges to an absorbing state in time polynomial in n, w.h.p., when any pair of agents is allowed to interact. When the interaction graph is a star, even the Rock-Paper-Scissors protocol requires exponential time to converge. We study threshold effects with three and more species under interactions between any pair of agents. We prove that the Rock-Paper-Scissors protocol reaches each of its three possible absorbing states with almost equal probability, starting from any configuration satisfying some sub-linear lower bound on the initial size of each species. Thus Rock-Paper-Scissors is a realization of “coin-flip consensus” in a distributed system.