Fixation time in evolutionary graphs: A mean-field approach

Hajihashemi, Mahdi; Samani, Keivan Aghababaei

doi:10.1103/physreve.99.042304

“…On more complicated networks, the probability of adding or removing a mutant depends on the configuration of existing mutants. For some of these networks, however, the transition probabilities can be accurately estimated using a mean-field approximation [19,23,25,26]. Then, to a good approximation, the results below apply to such networks as well.…”

Section: General Theory For Birth-death Markov Processesmentioning

confidence: 99%

“…(7) produces products of (b i + d i ) that arise in linear combinations determined by the visit numbers V ij . Therefore, the cumulants of the fixation time have the general form To the best of our knowledge this representation of the fixation-time cumulants has not been previously derived, although a similar approach was recently used to compute mean fixation times for evolutionary dynamics on complex networks [19]. This expression is equivalent to the well-known recurrence relations for absorption-time moments of birth-death processes [21,35] but is easier to handle asymptotically, and can be useful even without explicit expressions for w n i1i2•••in (r, N ).…”

Section: B Analytical Cumulant Calculation: Visit Statisticsmentioning

confidence: 99%

“…Of these quantities, the mean is the best understood. Numerical and analytical results exist for mean fixation times on both deterministic [4,6,[11][12][13][14][15][16][17] and random [16][17][18][19] networks. Yet although mean fixation times are important to study, the information they provide can be misleading, because fixation-time distributions tend to be broad and skewed and hence are not well characterized by their means alone [11,[20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

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Fitness dependence of the fixation-time distribution for evolutionary dynamics on graphs

Hathcock

¹

,

Strogatz

²

2019

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Evolutionary graph theory models the effects of natural selection and random drift on structured populations of competing mutant and non-mutant individuals. Recent studies have found that fixation times in such systems often have right-skewed distributions. Little is known, however, about how these distributions and their skew depend on mutant fitness. Here we calculate the fitness dependence of the fixation-time distribution for the Moran Birth-death process in populations modeled by two extreme networks: the complete graph and the one-dimensional ring lattice, obtaining exact solutions in the limit of large network size. We find that with non-neutral fitness, the Moran process on the ring has normally distributed fixation times, independent of the relative fitness of mutants and non-mutants. In contrast, on the complete graph, the fixation-time distribution is a fitness-weighted convolution of two Gumbel distributions. When fitness is neutral the fixationtime distribution jumps discontinuously and becomes highly skewed on both the complete graph and the ring. Even on these simple networks, the fixation-time distribution exhibits rich fitness dependence, with discontinuities and regions of universality. Extensions of our results to two-fitness Moran models, times to partial fixation, and evolution on random networks are discussed.

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“…To the best of our knowledge this representation of the fixation-time cumulants has not been previously derived, although a similar approach was recently used to compute mean fixation times for evolutionary dynamics on complex networks [19]. This expression is equivalent to the well-known recurrence relations for absorption-time moments of birth-death processes [21,35] but is easier to handle asymptotically, and can be useful even without explicit expressions for w n i1i2···in (r, N ).…”

Section: B Analytical Cumulant Calculation: Visit Statisticsmentioning

confidence: 99%

Fitness dependence of the fixation-time distribution for evolutionary dynamics on graphs

Hathcock

¹

,

Strogatz

²

2018

Preprint

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Evolutionary graph theory models the effects of natural selection and random drift on structured populations of competing mutant and non-mutant individuals. Recent studies have found that fixation times in such systems often have right-skewed distributions. Little is known, however, about how these distributions and their skew depend on mutant fitness. Here we calculate the fitness dependence of the fixation-time distribution for the Moran Birth-death process in populations modeled by two extreme networks: the complete graph and the one-dimensional ring lattice, obtaining exact solutions in the limit of large network size. We find that with non-neutral fitness, the Moran process on the ring has normally distributed fixation times, independent of the relative fitness of mutants and non-mutants. In contrast, on the complete graph, the fixation-time distribution is a fitness-weighted convolution of two Gumbel distributions. When fitness is neutral the fixationtime distribution jumps discontinuously and becomes highly skewed on both the complete graph and the ring. Even on these simple networks, the fixation-time distribution exhibits rich fitness dependence, with discontinuities and regions of universality. Extensions of our results to two-fitness Moran models, times to partial fixation, and evolution on random networks are discussed.

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“…Systemic risk: First, the simulations show that there is a vast variety of phenomena with random connections that arbitrarily act as robust amplifiers for the failure initialization. Many of those original properties are structurally simple (specifically, there are certain subset of vertices that we could call a small world because of their topology) but also strong (due to the influence based on those connections), and these could be realizable in other network structures such as regular and cycle in a fixation time of mutants [38].…”

Section: Discussionmentioning

confidence: 99%

Network and Agent Dynamics with Evolving Protection against Systemic Risk

Park

¹

2020

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The dynamics of protection processes has been a fundamental challenge in systemic risk analysis. The conceptual principle and methodological techniques behind the mechanisms involved (in such dynamics) have been harder to grasp than researchers understood them to be. In this paper, we show how to construct a large variety of behaviors by applying a simple algorithm to networked agents, which could, conceivably, offer a straightforward way out of the complexity. The model starts with the probability that systemic risk spreads. Even in a very random social structure, the propagation of risk is guaranteed by an arbitrary network property of a set of elements. Despite intensive systemic risk, the potential of the absence of failure could also be driven when there has been a strong investment in protection through a heuristically evolved protection level. It is very interesting to discover that many applications are still seeking the mechanisms through which networked individuals build many of these protection processes or mechanisms based on fitness due to evolutionary drift. At the same time, this approach could be useful for researchers and those who need to use protection dynamics to guard against systemic risk under intrinsic randomness in artificial circumstances.

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Fixation time in evolutionary graphs: A mean-field approach

Cited by 17 publications

References 38 publications

Fitness dependence of the fixation-time distribution for evolutionary dynamics on graphs

Fitness dependence of the fixation-time distribution for evolutionary dynamics on graphs

Fitness dependence of the fixation-time distribution for evolutionary dynamics on graphs

Network and Agent Dynamics with Evolving Protection against Systemic Risk

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