The Moran process is a randomised algorithm that models the spread of genetic mutations through graphs. If the graph is connected, the process eventually reaches "fixation", where every vertex is a mutant, or "extinction", where no vertex is a mutant. Our main result is an almost-tight bound on the expected running time of the algorithm. For all epsilon > 0, we show that the expected running time on an n-vertex graph is o(n^(3+epsilon)). In fact, we show that it is at most n^3 * exp(O((log log n)^3)) and that there is a family of graphs where it is Omega(n^3). In the course of proving our main result, we also establish a phase transition in the probability of fixation, depending on the fitness parameter r of the mutation. We show that no similar phase transition occurs for digraphs, where it is already known that the expected running time can also be exponential. Finally, we give an improved FPRAS for approximating the probability of fixation. Its running time is independent of the size of the graph when the maximum degree is bounded.
In , we alluded to the fact that the problem of finding the strongest possible amplifier had essentially been solved. Any directed graph has extinction probability @math @cite_15 , which is tight up to a polylogarithmic factor @cite_14 , and any undirected graph has extinction probability @math , which is tight up to a constant factor @cite_15 . In fact, the results of @cite_15 generalise to sparse graphs; any @math -edge undirected graph has extinction probability @math , which is also tight up to a constant factor. (See also Giakkoupis @cite_9 for a lower bound of @math in the dense undirected setting, which is proved to be tight to within a factor of @math .)
@cite_9 @cite_15 @cite_14
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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. The Moran process, as studied by Lieberman, Hauert, and Nowak, is a randomised algorithm modelling the spread of genetic mutations in populations. The algorithm runs on an underlying graph where individuals correspond to vertices. Initially, one vertex (chosen uniformly at random) possesses a mutation, with fitness r > 1. All other individuals have fitness 1. During each step of the algorithm, an individual is chosen with probability proportional to its fitness, and its state (mutant or nonmutant) is passed on to an out-neighbour which is chosen uniformly at random. If the underlying graph is strongly connected, then the algorithm will eventually reach fixation, in which all individuals are mutants, or extinction, in which no individuals are mutants. An infinite family of directed graphs is said to be strongly amplifying if, for every r > 1, the extinction probability tends to 0 as the number of vertices increases. A formal definition is provided in the article. Strong amplification is a rather surprising property—it means that in such graphs, the fixation probability of a uniformly placed initial mutant tends to 1 even though the initial mutant only has a fixed selective advantage of r > 1 (independently of n). The name “strongly amplifying” comes from the fact that this selective advantage is “amplified.” Strong amplifiers have received quite a bit of attention, and proposed two potentially strongly amplifying families—superstars and metafunnels. Heuristic arguments have been published, arguing that there are infinite families of superstars that are strongly amplifying. The same has been claimed for metafunnels. In this article, we give the first rigorous proof that there is an infinite family of directed graphs that is strongly amplifying. We call the graphs in the family “megastars.” When the algorithm is run on an n-vertex graph in this family, starting with a uniformly chosen mutant, the extinction probability is roughly n − 1 2 (up to logarithmic factors). We prove that all infinite families of superstars and metafunnels have larger extinction probabilities (as a function of n). Finally, we prove that our analysis of megastars is fairly tight—there is no infinite family of megastars such that the Moran algorithm gives a smaller extinction probability (up to logarithmic factors). Also, we provide a counterexample which clarifies the literature concerning the isothermal theorem of 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. Consider the random graph models G(n, #edges = M) and G(n, Prob(edge) = p) with M = M(n) = (1 + )n-1 3)n 2 and p = p(n) = (1 +An-1 3) n . For 1 > -1 define an i-component of a random graph as a component which has exactly I more edges than vertices. Call an i-component with I > 1 a complex component. For both models, we show that when A is constant, the expected number of complex components is bounded, almost surely (a.s.) each of these components (if any exist) has size of order n2 3, and the maximum value of I is bounded in probability. We prove that a.s. the largest suspended tree in each complex component has size of order n2 3, and deletion of all suspended trees results in a "smoothed" graph of size of order n 3, with the maximum vertex degree 3. The total number of branching vertices, i.e., of degree 3, is bounded in probability. Thus, each complex component is almost surely topologically equivalent to a 3-regular multigraph of a uniformly bounded size. Lengths of the shortest cycle and of the shortest path between two branching vertices of a smoothed graph are each of order n 3. We find a relatively simple integral formula for the limit distribution of the numbers of complex components, which implies, in particular, that all values of the "complexity spectrum" have positive limiting probabilities. We also answer questions raised by Erdos and Renyi back in 1960. It is proven that there exists p(A), the limiting planarity probability, with 0 < p(A) < 1, p(-oo) = 1, p(oo) = 0. In particular, G(n, M) (G(n, p), resp.) is almost surely nonplanar iff (M n 2)n-213 -* 00 ((np I)n-1 3) -+ oo, resp.).