We consider the classic Moran process modeling the spread of genetic mutations, as extended to structured populations by (Nature, 2005). In this process, individuals are the vertices of a connected graph G. Initially, there is a single mutant vertex, chosen uniformly at random. In each step, a random vertex is selected for reproduction with a probability proportional to its fitness: mutants have fitness r > 1, while non-mutants have fitness 1. The vertex chosen to reproduce places a copy of itself to a uniformly random neighbor in G, replacing the individual that was there. The process ends when the mutation either reaches fixation (i.e., all vertices are mutants), or gets extinct. The principal quantity of interest is the probability with which each of the two outcomes occurs. A problem that has received significant attention recently concerns the existence of families of graphs, called strong amplifiers of selection, for which the fixation probability tends to 1 as the order n of the graph increases, and the existence of strong suppressors of selection, for which this probability tends to 0. For the case of directed graphs, it is known that both strong amplifiers and suppressors exist. For the case of undirected graphs, however, the problem has remained open, and the general belief has been that neither strong amplifiers nor suppressors exist. In this paper we disprove this belief, by providing the first examples of such graphs. The strong amplifier we present has fixation probability 1 − ˜ O(n^ −1 3), and the strong suppressor has fixation probability ˜O(n^−1 4). Both graph constructions are surprisingly simple. We also prove a general upper bound of 1 − ˜Ω(n^−1 3) on the fixation probability of any undirected graph. Hence, our strong amplifier is existentially optimal.
Several works have been devoted to the numerical computation of the fixation probability on specific classes of graphs, such as instances of bipartite graphs and Erd "os-R 'enyi random graphs @cite_3 @cite_18 @cite_34 @cite_4 .
@cite_18 @cite_34 @cite_4 @cite_3
2139738313 2950361113 2130745630 2205121374
The problem of finding birth–death fixation probabilities for configurations of normal and mutants on an N -vertex graph is formulated in terms of a Markov process on the 2 N -dimensional state space of possible configurations. Upper and lower bounds on the fixation probability after any given number of iterations of the birth–death process are derived in terms of the transition matrix of this process. Consideration is then specialized to a family of graphs called circular flows, and we present a summation formula for the complete bipartite graph, giving the fixation probability for an arbitrary configuration of mutants in terms of a weighted sum of the single-vertex fixation probabilities. This also yields a closed-form solution for the fixation probability of bipartite graphs. Three entropy measures are introduced, providing information about graph structure. Finally, a number of examples are presented, illustrating cases of graphs that enhance or suppress fixation probability for fitness r >1 as well as graphs that enhance fixation probability for only a limited range of fitness. Results are compared with recent results reported in the literature, where a positive correlation is observed between vertex degree variance and fixation probability for undirected graphs. We show a similar correlation for directed graphs, with correlation not directly to fixation probability but to the difference between fixation probability for a given graph and a complete graph. Consider a collection of random variables attached to the vertices of a graph. The reconstruction problem requires to estimate one of them given far away' observations. Several theoretical results (and simple algorithms) are available when their joint probability distribution is Markov with respect to a tree. In this paper we consider the case of sequences of random graphs that converge locally to trees. In particular, we develop a sufficient condition for the tree and graph reconstruction problem to coincide. We apply such condition to colorings of random graphs. Further, we characterize the behavior of Ising models on such graphs, both with attractive and random interactions (respectively, ferromagnetic' and spin glass'). We study the existence of perfect matchings in suitably chosen induced subgraphs of random biregular bipartite graphs. We prove a result similar to a classical theorem of Erdos and Renyi about perfect matchings in random bipartite graphs. We also present an application to commutative graphs, a class of graphs that are featured in additive number theory. This work extends what is known so far for a basic model of evolutionary antagonism in undirected networks (graphs). More specifically, this work studies the generalized Moran process, as introduced by Lieberman, Hauert, and Nowak [Nature, 433:312-316, 2005], where the individuals of a population reside on the vertices of an undirected connected graph. The initial population has a single mutant of a fitness value r (typically r>1), residing at some vertex v of the graph, while every other vertex is initially occupied by an individual of fitness 1. At every step of this process, an individual (i.e. vertex) is randomly chosen for reproduction with probability proportional to its fitness, and then it places a copy of itself on a random neighbor, thus replacing the individual that was residing there. The main quantity of interest is the fixation probability, i.e. the probability that eventually the whole graph is occupied by descendants of the mutant. In this work we concentrate on the fixation probability when the mutant is initially on a specific vertex v, thus refining the older notion of which studied the fixation probability when the initial mutant is placed at a random vertex. We then aim at finding graphs that have many "strong starts" (or many "weak starts") for the mutant. Thus we introduce a parameterized notion of selective amplifiers (resp. selective suppressors) of evolution. We prove the existence of strong selective amplifiers (i.e. for h(n)=Θ(n) vertices v the fixation probability of v is at least @math for a function c(r) that depends only on r), and the existence of quite strong selective suppressors. Regarding the traditional notion of fixation probability from a random start, we provide strong upper and lower bounds: first we demonstrate the non-existence of "strong universal" amplifiers, and second we prove the Thermal Theorem which states that for any undirected graph, when the mutant starts at vertex v, the fixation probability at least @math . This theorem (which extends the "Isothermal Theorem" of for regular graphs) implies an almost tight lower bound for the usual notion of fixation probability. Our proof techniques are original and are based on new domination arguments which may be of general interest in Markov Processes that are of the general birth-death type.