Asymptotically optimal amplifiers for the Moran process

Goldberg, Leslie Ann; Lapinskas, John; Lengler, Johannes; Meier, Florian; Παναγιώτου, Κωνσταντίνος; Pfister, Pascal

doi:10.1016/j.tcs.2018.08.005

Cited by 18 publications

(28 citation statements)

References 18 publications

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“…For the Bd Moran process, various results on amplifiers exist. The Star graph is a prominent example of a graph that is a quadratic amplifier for any r > 1 [24,25,27,26,28] and there exist super amplifiers, that is, families of graphs that guarantee fixation in the limit of large population size, for any fixed r > 1 [16,29,30,31,32]. Furthermore, computer simulations on small populations have shown that many small graphs have amplifying properties [33,34,35].…”

Section: Questionsmentioning

confidence: 99%

Limits on amplifiers of natural selection under death-Birth updating

et al. 2020

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The fixation probability of a single mutant invading a population of residents is among the most widely-studied quantities in evolutionary dynamics. Amplifiers of natural selection are population structures that increase the fixation probability of advantageous mutants, compared to well-mixed populations. Extensive studies have shown that many amplifiers exist for the Birth-death Moran process, some of them substantially increasing the fixation probability or even guaranteeing fixation in the limit of large population size. On the other hand, no amplifiers are known for the death-Birth Moran process, and computer-assisted exhaustive searches have failed to discover amplification. In this work we resolve this disparity, by showing that any amplification under death-Birth updating is necessarily bounded and transient. Our boundedness result states that even if a population structure does amplify selection, the resulting fixation probability is close to that of the well-mixed population.

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Section: Questionsmentioning

confidence: 99%

Limits on amplifiers of natural selection under death-Birth updating

et al. 2020

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“…In Section 1.2, we alluded to the fact that the problem of finding the strongest possible amplifier had essentially been solved. Any directed graph has extinction probability Ω(n −1∕2 ) [13], which is tight up to a polylogarithmic factor [11], and any undirected graph has extinction probability Ω(n −1∕3 ), which is tight up to a constant factor [13]. In fact, the results of [13] generalize to sparse graphs; any m-edge undirected graph has extinction probability Ω(max{n −1∕3 , n∕m}), which is also tight up to a constant factor.…”

Section: Related Workmentioning

confidence: 95%

“…It is natural to ask: how strong can an n ‐vertex amplifier or suppressor be? For amplifiers this problem has essentially been solved for both directed and undirected graphs ; see Section 1.4 for details. For suppressors, much less is known; to our knowledge, the strongest family of (both directed and undirected) suppressors in the literature is due to Giakkoupis , and has fixation probability

O false(n^{prefix- 1 false/ 4} \log n false)

.…”

Section: Introductionmentioning

confidence: 99%

“…Any directed graph has extinction probability Ω( n −1/2 ) , which is tight up to a polylogarithmic factor , and any undirected graph has extinction probability Ω( n −1/3 ), which is tight up to a constant factor . In fact, the results of generalize to sparse graphs; any m ‐edge undirected graph has extinction probability

normalΩ false(\max false{n^{prefix- 1 false/ 3}, n false/ m false} false)

, which is also tight up to a constant factor. (See also Giakkoupis for a lower bound of

normalΩ false(n^{prefix- 1 false/ 3} {false(\log n false)}^{prefix- 4 false/ 3} false)

in the dense undirected setting, which is proved to be tight to within a factor of

{false(\log n false)}^{7 false/ 3}

.…”

Section: Introductionmentioning

confidence: 99%

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Phase transitions of the Moran process and algorithmic consequences

Goldberg

Lapinskas

Richerby

2019

Random Struct Algorithms

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The Moran process is a random process that models the spread of genetic mutations through graphs. On connected graphs, the process eventually reaches “fixation,” where all vertices are mutants, or “extinction,” where none are. Our main result is an almost‐tight upper bound on expected absorption time. For all ϵ>0, we show that the expected absorption time on an n‐vertex graph is o(n3+ϵ). Specifically, it is at most n3eOfalse(false(loglognfalse)3false), and there is a family of graphs where it is Ω(n3). In proving this, we establish a phase transition in the probability of fixation, depending on the mutants' fitness r. We show that no similar phase transition occurs for digraphs, where it is already known that the expected absorption time can be exponential. Finally, we give an improved fully polynomial randomized approximation scheme (FPRAS) for approximating the probability of fixation. On degree‐bounded graphs where some basic properties are given, its running time is independent of the number of vertices.

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“…families of graphs with fixation probability tending to 1 or to 0 respectively as the order of the graph tends to infinity and for r > 1. Galanis et al [4] find an infinite family of strongly-amplifying directed graphs, namely the "megastar" with fixation probability 1 − O(n −1/2 log 23 n), which was later proved to be optimal up to logarithmic factors [6].…”

Section: Previous Workmentioning

confidence: 99%

Mutants and Residents with Different Connection Graphs in the Moran Process

Melissourgos

Nikoletseas

Raptopoulos

et al. 2018

LATIN 2018: Theoretical Informatics

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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.

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Asymptotically optimal amplifiers for the Moran process

Cited by 18 publications

References 18 publications

Limits on amplifiers of natural selection under death-Birth updating

Limits on amplifiers of natural selection under death-Birth updating

Phase transitions of the Moran process and algorithmic consequences

Mutants and Residents with Different Connection Graphs in the Moran Process

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