Martingales and the fixation probability of high-dimensional evolutionary graphs

Monk, Travis

doi:10.1016/j.jtbi.2018.04.039

Cited by 20 publications

(41 citation statements)

References 29 publications

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“…How does population structure affect evolutionary dynamics? This question is at the center of evolutionary graph theory, since its introduction by Lieberman et al 1 and remains a flourishing research topic 2–13 . While the bulk of the associated work is theoretical at this stage, there are many potential applications: In ecology, river systems often have tree-like structures and the spread of new ecotypes or species happens along the branches of this tree.…”

Section: Introductionmentioning

confidence: 99%

Exploring and mapping the universe of evolutionary graphs identifies structural properties affecting fixation probability and time

2019

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Population structure can be modeled by evolutionary graphs, which can have a substantial influence on the fate of mutants. Individuals are located on the nodes of these graphs, competing to take over the graph via the links. Applications for this framework range from the ecology of river systems and cancer initiation in colonic crypts to biotechnological search for optimal mutations. In all these applications, both the probability of fixation and the associated time are of interest. We study this problem for all undirected and unweighted graphs up to a certain size. We devise a genetic algorithm to find graphs with high or low fixation probability and short or long fixation time and study their structure searching for common themes. Our work unravels structural properties that maximize or minimize fixation probability and time, which allows us to contribute to a first map of the universe of evolutionary graphs.

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Section: Introductionmentioning

confidence: 99%

Exploring and mapping the universe of evolutionary graphs identifies structural properties affecting fixation probability and time

2019

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“…We can often 24 calculate absorption (i.e. fixation) probabilities [8,13] and/or times [17,18] from it. 25 Abraham Wald identified a powerful martingale for stochastic processes whose steps 26 are independent and identically distributed [17,18].…”

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“…Our CFs for the 35 number of mutant population size changes are novel, clean, and exact results. More 36 generally, Wald's methodology demonstrates an elegant approach to investigate classic 37 problems of evolutionary models [8,13,22]. 38 Results 39 Fig.…”

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confidence: 99%

“…If we consider more complicated evolutionary stochastic 249 processes, e.g. an evolutionary graph with many partitions [8,11,38], Markov chains 250 quickly become an infeasible method of analysis. Martingales circumvent these issues 251 because they are conservation statements, so we do not need to evaluate recursions over 252 state space.…”

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confidence: 99%

“…We can modify the process such that transition probabilities are 269 state-independent, as we have done here by discarding time steps where the mutant 270 population size remains unchanged. Alternatively, we can show that the expectation of 271 some state-dependent quantity always equals 1 (a product martingale) or 0 (a sum 272 martingale), regardless of the state [8,13]. Finding such a quantity can be difficult, in…”

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Wald’s martingale and the Moran process

Monk

Schaik

2020

Preprint

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Many models of evolution are stochastic processes, where some quantity of interest fluctuates randomly in time. One classic example is the Moran birth-death process, where that quantity is the number of mutants in a population. In such processes we are often interested in their absorption (i.e. fixation) probabilities, and the conditional distributions of absorption time. Those conditional time distributions can be very difficult to calculate, even for relatively simple processes like the Moran birth-death model. Instead of considering the time to absorption, we consider a closely-related quantity: the number of mutant population size changes before absorption. We use Wald's martingale to obtain the conditional characteristic functions of that quantity in the Moran process. Our expressions are novel, analytical, and exact. The parameter dependence of the characteristic functions is explicit, so it is easy to explore their properties in parameter space. We also use them to approximate the conditional characteristic functions of absorption time. We state the conditions under which that approximation is particularly accurate. Martingales are an elegant framework to solve principal problems of evolutionary stochastic processes. They do not require us to evaluate recursion relations, so we can quickly and tractably obtain absorption probabilities and times of evolutionary stochastic processes.February 18, 2020 1/21The Moran process is a probabilistic birth-death model of evolution. A mutant is introduced to an indigenous population, and we randomly choose organisms to live or die on subsequent time steps. Our goals are to calculate the probabilities that the mutant eventually dominates the population or goes extinct, and the distribution of time it requires to do so. The conditional distributions of time are difficult to obtain for the Moran process, so we consider a slightly different but related problem. We instead calculate the conditional distributions of the number of times that the mutant population size changes before it dominates the population or goes extinct. We use a martingale identified by Abraham Wald to obtain elegant and exact expressions for those distributions. We then use them to approximate conditional time distributions, and we show when that approximation is accurate. Our analysis outlines the basic concepts martingales and demonstrates why they are a formidable tool for studying probabilistic evolutionary models such as the Moran process. Introduction 1 The Moran birth-death process is a classic model of evolution [1, 2]. It considers a 2 population of two species that we call mutants and indigenous. On sequential time 3 steps, one organism is chosen to reproduce from either species. The newborn is the 4 same species as its parent, and it replaces another organism in the population uniformly 5 at random. The total population size then remains constant, but the numbers of mutant 6 and indigenous organisms in it are stochastic in time. The mutants will drive the 7 indigenous organisms to extinctio...

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Spectral analysis of transient amplifiers for death–birth updating constructed from regular graphs

Richter

2021

J. Math. Biol.

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A central question of evolutionary dynamics on graphs is whether or not a mutation introduced in a population of residents survives and eventually even spreads to the whole population, or becomes extinct. The outcome naturally depends on the fitness of the mutant and the rules by which mutants and residents may propagate on the network, but arguably the most determining factor is the network structure. Some structured networks are transient amplifiers. They increase for a certain fitness range the fixation probability of beneficial mutations as compared to a well-mixed population. We study a perturbation method for identifying transient amplifiers for death–birth updating. The method involves calculating the coalescence times of random walks on graphs and finding the vertex with the largest remeeting time. If the graph is perturbed by removing an edge from this vertex, there is a certain likelihood that the resulting perturbed graph is a transient amplifier. We test all pairwise nonisomorphic regular graphs up to a certain order and thus cover the whole structural range expressible by these graphs. For cubic and quartic regular graphs we find a sufficiently large number of transient amplifiers. For these networks we carry out a spectral analysis and show that the graphs from which transient amplifiers can be constructed share certain structural properties. Identifying spectral and structural properties may promote finding and designing such networks.

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Martingales and the fixation probability of high-dimensional evolutionary graphs

Cited by 20 publications

References 29 publications

Exploring and mapping the universe of evolutionary graphs identifies structural properties affecting fixation probability and time

Exploring and mapping the universe of evolutionary graphs identifies structural properties affecting fixation probability and time

Wald’s martingale and the Moran process

Spectral analysis of transient amplifiers for death–birth updating constructed from regular graphs

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