20150101
On convergence and threshold properties of discrete lotkavolterra population protocols
Publication
Publication
In this work we focus on a natural class of population protocols whose dynamics are modeled by the discrete version of LotkaVolterra equations with no linear term. In such protocols, when an agent a of type (species) i interacts with an agent b of type (species) j with a as the initiator, then b’s type becomes i with probability Pij. In such an interaction, we think of a as the predator, b as the prey, and the type of the prey is either converted to that of the predator or stays as is. Such protocols capture the dynamics of some opinion spreading models and generalize the wellknown RockPaperScissors discrete dynamics. We consider the pairwise interactions among agents that are scheduled uniformly at random. We start by considering the convergence time and show that any LotkaVolterratype protocol on an nagent population converges to some absorbing state in time polynomial in n, w. h. p., when any pair of agents is allowed to interact. By contrast, when the interaction graph is a star, there exist protocols of the considered type, such as RockPaperScissors, which require exponential time to converge. We then study threshold effects exhibited by LotkaVolterratype protocols with 3 and more species under interactions between any pair of agents. We present a simple 4type protocol in which the probability difference of reaching the two possible absorbing states is strongly amplified by the ratio of the initial populations of the two other types, which are transient, but “control” convergence. We then prove that the RockPaperScissors protocol reaches each of its three possible absorbing states with almost equal probability, starting from any configuration satisfying some sublinear lower bound on the initial size of each species. That is, RockPaperScissors is a realization of a “coinflip consensus” in a distributed system. Some of our techniques may be of independent value.
Additional Metadata  

Persistent URL  dx.doi.org/10.1007/9783662476727_32 
Citation 
Czyzowicz, J. (Jurek), Ģasieniec, L. (Leszek), Kosowski, A. (Adrian), Kranakis, E, Spirakis, P.G. (Paul G.), & Uznański, P. (Przemysław). (2015). On convergence and threshold properties of discrete lotkavolterra population protocols. doi:10.1007/9783662476727_32
