Mutants and Residents with Different Connection Graphs in the Moran Process

Abstract

The Moran process, as studied by Lieberman, Hauert and Nowak [10], 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 (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 [10] by assuming that different types of individuals perceive the population through different graphs defined on the same vertex set, namely GR = (V, ER) for residents and GM = (V, EM ) for mutants. In this model, we study the fixation probability, namely the probability that eventually only mutants remain in the population, for various pairs of graphs. In particular, in the first part of the paper, we examine how known results from the original single-graph model of [10] can be transferred to our 2-graph model. In that direction, by using a Markov chain abstraction, we provide a generalization of the Isothermal Theorem of [10], that gives sufficient conditions for a pair of graphs to have fixation probability equal to the fixation probability of a pair of cliques; this corresponds to the absorption probability of a birth-death process with forward bias r. In the second part of the paper, 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. We give evidence that the clique is the most beneficial graph for both players, by proving bounds on the fixation probability when one of the two graphs is complete and the other graph belongs to various natural graph classes. In the final part of the paper, we examine the possibility of efficient approximation of the fixation probability. Interestingly, we show that This work can be found at the proceedings of LATIN 2018 [11]. The work of this author was partially supported by the ERC Project ALGAME. arXiv:1710.07365v2 [q-bio.PE] 23 May 2018 2 T. Melissourgos, S. Nikoletseas, C. Raptopoulos and P. Spirakis there is a pair of graphs for which the fixation probability is exponentially small. This implies that the fixation probability in the general case of an arbitrary pair of graphs cannot be approximated via a method similar to [2]. Nevertheless, we prove that, in the special case when the mutant graph is complete, an efficient approximation of the fixation probability is possible through an FPRAS which we describe.

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

“…As it is apparent by the Markov chain abstraction, the fixation probability f (X) starting from a mutant set X is the absorption probability for the absorbing state V [22]. This absorption probability can be calculated from the set of equations ( 1), or after eliminating self-loops, from the equations (2). For ease of presentation, let us focus on the latter, and also denote by p X+j X and p X−i X the transition probabilities which are coefficients of f (X + j) and f (X − i) in (2).…”

“…This absorption probability can be calculated from the set of equations ( 1), or after eliminating self-loops, from the equations (2). For ease of presentation, let us focus on the latter, and also denote by p X+j X and p X−i X the transition probabilities which are coefficients of f (X + j) and f (X − i) in (2). Let M = (M (t)) t≥0 be a Moran process, and for each k ∈ {0, 1, .…”

Phase transitions of the Moran process and algorithmic consequences

“…Proof. Let U = V, so that s = (log r n) 1∕3 and the quantity C(log n) 1∕3 , from the failure probability in the statement of the theorem, is equal to C(log r) 1∕3 s. We will choose C to be small enough (as a function of r) so that, when n is sufficiently large, the final inequalities of (20) and (21)…”

“…Some versions of the process allow edge weights, which is equivalent to allowing multiple edges in the graph, and others determine the fitness of a vertex partly or fully by game payoffs with its neighbors. More recently, a variant has been proposed in which the mutants and nonmutants interact along different graphs. In this paper, we consider the original process of Lieberman et al , with a single, simple, unweighted graph and mutants with fixed fitness r .…”

Phase Transitions of the Moran Process and Algorithmic Consequences

“…Some versions of the process allow edge weights, which is equivalent to allowing multiple edges in the graph, and others determine the fitness of a vertex partly or fully by game payoffs with its neighbours. More recently, a variant has been proposed [20] in which the mutants and non-mutants interact along different graphs. In this paper, we consider the original process of Lieberman et al [18], with a single, simple, unweighted graph and mutants with fixed fitness r.…”