We study the Moran process as adapted by Lieberman, Hauert and Nowak. This is a model of an evolving population on a graph where certain individuals, called "mutants" have fitness r and other individuals, called "non-mutants" have fitness 1. We focus on the situation where the mutation is advantageous, in the sense that r > 1. A family of directed graphs is said to be strongly amplifying if the extinction probability tends to 0 when the Moran process is run on graphs in this family. The most-amplifying known family of directed graphs is the family of megastars of We show that this family is optimal, up to logarithmic factors, since every strongly-connected n-vertex digraph has extinction probability Omega(n^(-1 2)). Next, we show that there is an infinite family of undirected graphs, called dense incubators, whose extinction probability is O(n^(-1 3)). We show that this is optimal, up to constant factors. Finally, we introduce sparse incubators, for varying edge density, and show that the extinction probability of these graphs is O(n m), where m is the number of edges. Again, we show that this is optimal, up to constant factors.
The best-known lower bounds on the extinction probability of connected undirected graphs are in @cite_8 @cite_13 . Theorem 1 of @cite_9 shows that there is a constant @math such that for every @math the extinction probability is at least @math .
@cite_9 @cite_13 @cite_8
2552612746 2012430523 2056986231 2037549896
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. Let P be a Poisson process of intensity one in a square Sn of area n. For a fixed integer k, join every point of P to its k nearest neighbours, creating an undirected random geometric graph Gn,k. We prove that there exists a critical constant ccrit such that for c ccrit, Gn,⌊clog n⌋ is connected with probability tending to 1 as n → ∞. This answers a question posed by the authors in [1]. Let P be a Poisson process of intensity one in a square Sn of area n. For a fixed integer k, we join every point of P to its k nearest neighbours, creating an undirected random geometric graph GSn,k = Gn,k in which every vertex has degree at least k. The connectivity of these graphs was studied by the present authors in [1]. It is not hard to see that Gn,k becomes connected around k = �(log n), and we proved in [1] that if k(n) ≤ 0.3043log n then the probability that Gn,k(n) is connected tends to zero as n → ∞, while if k(n) ≥ 0.5139log n then the probability that Gn,k(n) is connected tends to one as n → ∞. However, we were unable to prove the natural conjecture that there exists a critical constant ccrit such that for c ccrit, P(Gn,⌊clog n⌋ is connected) → 1 as n → ∞. In this paper we prove this conjecture. Let P be a Poisson process of intensity one in a square S n of area n. We construct a random geometric graph G n , k by joining each point of P to its k ≡ k(n) nearest neighbours. Recently, Xue and Kumar proved that if k ≤ 0.074 log n then the probability that G n , k is connected tends to 0 as n → ∞ while, if k ≥ 5.1774 log n, then the probability that G n , k is connected tends to 1 as n → ∞. They conjectured that the threshold for connectivity is k = (1 + o(1)) log n. In this paper we improve these lower and upper bounds to 0.3043 log n and 0.5139 log n, respectively, disproving this conjecture. We also establish lower and upper bounds of 0.7209 log n and 0.9967 log n for the directed version of this problem. A related question concerns coverage. With G n , k as above, we surround each vertex by the smallest (closed) disc containing its k nearest neighbours. We prove that if k ≤ 0.7209 log n then the probability that these discs cover S n tends to 0 as n → ∞ while, if k > 0.9967 log n, then the probability that the discs cover S n tends to 1 as n → ∞. ean distance is at most r, for some prescribed r. We show that monotone properties for this class of graphs have sharp thresholds by reducing the problem to bounding the bottleneck matching on two sets of n points distributed uniformly in [0, 1] d . We present upper bounds on the threshold width, and show that our bound is sharp for d = 1 and at most a sublogarithmic factor away for d ≥ 2. Interestingly, the threshold width is much sharper for random geometric graphs than for Bernoulli random graphs. Further, a random geometric graph is shown to be a subgraph, with high probability, of another independently drawn random geometric graph with a slightly larger radius; this property is shown to have no analogue for Bernoulli random graphs.