Resource competition and survival for random walkers on complex networks

Abstract

We consider the problem of $W$ random walkers on a complex network where each walker requires an amount $θ$ of resource at each time to survive. Resource is stochastically produced at each vertex of the network with a specified distribution. The survival time of the system of walkers is defined as the time required for all $W$ walkers to die. Using numerical simulations, we show that Erd{\H o}s-Rényi networks a have higher mean survival time than Scale-free networks as expected, but also that uniform distribution production leads to higher mean survival time than exponential distribution with the same mean. We derive an exact expression for the survival-time distribution under the assumption of independent walker lifespans, and use it to show that scale-free network topology produces strong dependence in the lifespans but homogeneous topology does not. We moreover demonstrate how the survival time can be substantially increased without changing the resource production or network topology by appropriately biasing the movement of the walkers towards low degree vertices on Scale-free networks, but has no effect on Erd{\H o}s-R{é}nyi networks.