An extension of the Moran process using type-specific connection graphs

The Moran process, as studied by Lieberman, Hauert and Nowak (2005) [1], is a birth-death process that models the spread of mutations in two-type populations (residents-mutants) whose structure is defined by a digraph. The process' central notion is the probability that a randomly placed mutant will occupy the whole vertex set (fixation probability). We extend this model by considering type-specific graphs, and consequently present results on the fundamental problems related to the fixation probability and its computation. Finally, we view the resident-mutant competing forces as players that choose digraphs and indicate that the mutant's complete graph is a dominant strategy

An extension of the Moran process using type-specific connection graphs

The Moran process, as studied by Lieberman et al. [L05], is a stochastic process modeling the spread of genetic mutations in populations. In this process, agents of a two-type population (i.e. mutants and residents) are associated with the vertices of a graph. Initially, only one vertex chosen u.a.r. is a mutant, with fitness $r > 0$, while all other individuals are residents, with fitness $1$. In every step, an individual is chosen with probability proportional to its fitness, and its state (mutant or resident) is passed on to a neighbor which is chosen u.a.r. In this paper, we introduce and study for the first time a generalization of the model of [L05] by assuming that different types of individuals perceive the population through different graphs, namely $G_R(V,E_R)$ for residents and $G_M(V,E_M)$ for mutants. In this model, we study the fixation probability, i.e. the probability that eventually only mutants remain in the population, for various pairs of graphs. First, we transfer known results from the original single-graph model of [L05] to our 2-graph model. Among them, we provide a generalization of the Isothermal Theorem of [L05], that gives sufficient conditions for a pair of graphs to have the same fixation probability as a pair of cliques. Next, we give a 2-player strategic game view of the process where player payoffs correspond to fixation and/or extinction probabilities. In this setting, we attempt to identify best responses for each player and give evidence that the clique is the most beneficial graph for both players. Finally, we examine the possibility of efficient approximation of the fixation probability. We show that the fixation probability in the general case of an arbitrary pair of graphs cannot be approximated via a method similar to [D14]. Nevertheless, we provide a FPRAS for the special case where the mutant graph is complete.Comment: This work can be found at the proceedings of LATIN 201