  The Moran process, as studied by [Lieberman, E., Hauert, C. and Nowak, M.
Evolutionary dynamics on graphs. Nature 433, pp. 312-316 (2005)], 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
uniformly at random 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 uniformly at random. In
this paper, we introduce and study a generalization of the model of Lieberman
et al. 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 Lieberman et al. to our 2-graph model. Among them, we
provide a generalization of the Isothermal Theorem of Lieberman et al., 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 and provide a FPRAS for the special case where the mutant
graph is complete.
