Martingales and the fixation time of evolutionary graphs with arbitrary dimensionality

Monk, Travis; Schaik, André van

doi:10.1098/rsos.220011

Cited by 4 publications

(9 citation statements)

References 69 publications

(185 reference statements)

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“…The prospects of engineering a population structure that can optimize the chances to evolve certain mutations or to observe evolved population structures that minimize the evolution of mutations seem exciting, but these applications call for an extension of the field of evolutionary graph theory: Most applications implicitly assume that each node is a small population, and not all results carry over from graphs of individuals to graphs of subpopulations ( 38 – 41 ). In addition, the field has focused so far on fixation probability and fixation time ( 42 – 49 ).…”

Section: Discussionmentioning

confidence: 99%

Suppressors of fixation can increase average fitness beyond amplifiers of selection

Sharma

Traulsen

2022

Proc. Natl. Acad. Sci. U.S.A.

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Evolutionary dynamics on graphs has remarkable features: For example, it has been shown that amplifiers of selection exist that—compared to an unstructured population—increase the fixation probability of advantageous mutations, while they decrease the fixation probability of disadvantageous mutations. So far, the theoretical literature has focused on the case of a single mutant entering a graph-structured population, asking how the graph affects the probability that a mutant takes over a population and the time until this typically happens. For continuously evolving systems, the more relevant case is that mutants constantly arise in an evolving population. Typically, such mutations occur with a small probability during reproduction events. We thus focus on the low mutation rate limit. The probability distribution for the fitness in this process converges to a steady state at long times. Intuitively, amplifiers of selection are expected to increase the population’s mean fitness in the steady state. Similarly, suppressors of selection are expected to decrease the population’s mean fitness in the steady state. However, we show that another set of graphs, called suppressors of fixation, can attain the highest population mean fitness. The key reason behind this is their ability to efficiently reject deleterious mutants. This illustrates the importance of the deleterious mutant regime for the long-term evolutionary dynamics, something that seems to have been overlooked in the literature so far.

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Section: Discussionmentioning

confidence: 99%

Suppressors of fixation can increase average fitness beyond amplifiers of selection

Sharma

Traulsen

2022

Proc. Natl. Acad. Sci. U.S.A.

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“…, (19) which in the limit N → ∞ approaches unity. Since the fixation is opposite to the mutation elimination [Π j ≈ 1 − P j (elimination) in this limit], one can see that it is more probable to remove the mutation from the central cell, while it is much less probable to eliminate the mutation from the branched cell.…”

Section: Evolutionary Dynamics On Star Networkmentioning

confidence: 95%

“…the fixation times in these systems are always larger than the fixation times for similar-size homogeneous well-mixed populations [33]. Although the fixation processes for inhomogeneous populations have been intensively studied in recent years [3,5,6,18,19,25,32,34], there is still no clear understanding on the microscopic origin of selection amplifications, the connections to the underlying network topology, and the correlations between the fixation probabilities and the fixation times.…”

Section: Introductionmentioning

confidence: 99%

Theoretical understanding of evolutionary dynamics on inhomogeneous networks

et al. 2023

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Evolution is the main feature of all biological systems that allows populations to change their characteristics over successive generations. A powerful approach to understand evolutionary dynamics is to investigate fixation probabilities and fixation times of novel mutations on networks that mimic biological populations. It is now well established that the structure of such networks can have dramatic effects on evolutionary dynamics. In particular, there are population structures that might amplify the fixation probabilities while simultaneously delaying the fixation events. However, the microscopic origins of such complex evolutionary dynamics remain not well understood. We present here a theoretical investigation of the microscopic mechanisms of mutation fixation processes on inhomogeneous networks. It views evolutionary dynamics as a set of stochastic transitions between discrete states specified by different numbers of mutated cells. By specifically considering star networks, we obtain a comprehensive description of evolutionary dynamics. Our approach allows us to employ physics-inspired free-energy landscape arguments to explain the observed trends in fixation times and fixation probabilities, providing a better microscopic understanding of evolutionary dynamics in complex systems.

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“…royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 21: 20230594 it numerically intractable (except for estimates of the lower bound of the sojourn time on graphs [37] and the fixation time of k-partite street graphs [38] with the help of Martingales and other special graphs). (ii) The graph with six nodes leads to 112 graphs that are not isomorphic, which is not few, and allows us to obtain exact results.…”

Section: Network and Joint Degree Distributionmentioning

confidence: 99%

“…The initial position of the mutant on the graph is critical on graphs like star and k -partite street graph [37,38]. Here, we assume that any node has an equal probability of becoming a mutant.…”

Section: Introductionmentioning

confidence: 99%

The speed of neutral evolution on graphs

Gao,

Liu,

2024

J. R. Soc. Interface.

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The speed of evolution on structured populations is crucial for biological and social systems. The likelihood of invasion is key for evolutionary stability. But it makes little sense if it takes long. It is far from known what population structure slows down evolution. We investigate the absorption time of a single neutral mutant for all the 112 non-isomorphic undirected graphs of size 6. We find that about three-quarters of the graphs have an absorption time close to that of the complete graph, less than one-third are accelerators, and more than two-thirds are decelerators. Surprisingly, determining whether a graph has a long absorption time is too complicated to be captured by the joint degree distribution. Via the largest sojourn time, we find that echo-chamber-like graphs, which consist of two homogeneous graphs connected by few sparse links, are likely to slow down absorption. These results are robust for large graphs, mutation patterns as well as evolutionary processes. This work serves as a benchmark for timing evolution with complex interactions, and fosters the understanding of polarization in opinion formation.

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Martingales and the fixation time of evolutionary graphs with arbitrary dimensionality

Cited by 4 publications

References 69 publications

Suppressors of fixation can increase average fitness beyond amplifiers of selection

Suppressors of fixation can increase average fitness beyond amplifiers of selection

Theoretical understanding of evolutionary dynamics on inhomogeneous networks

The speed of neutral evolution on graphs

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