Analytical calculation of average fixation time in evolutionary graphs

Askari, Marziyeh; Samani, Keivan Aghababaei

doi:10.1103/physreve.92.042707

Cited by 35 publications

(45 citation statements)

References 12 publications

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“…Which is in agreement with the results obtained by other methods [4,8,11]. Now we examine this method on a random (Erdös-Rényi) network [13] and calculate the fixation time.…”

supporting

confidence: 86%

“…Our method is applicable for a general absorbing Markov chain with arbitrary number of absorbing states [12]. Applying this method to complete and circle graphs shows complete agreement with the results obtained from recursive equation methods [8,11]. Generally for graphs whose transition matrices satisfy the condition λ i = rµ i , obtaining inverse of tridiagonal matrix (I −Q) is straightforward.…”

supporting

confidence: 55%

“…This can be easily seen for the present Markov chain using the recursive method of Ref. [8], namely by using the recursive relation…”

mentioning

confidence: 91%

“…Fixation probability is the probability for a single mutant to take over the whole population and fixation time is the average (conditional) time needed for this result. Both of these quantities are investigated by many researchers [4][5][6][7][8][9]. Obtaining fixation time is more challenging than fixation probability.…”

mentioning

confidence: 99%

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Fixation time in evolutionary graphs: a mean field approach

jazi

Samani

2018

Preprint

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A novel analytical method is proposed for calculation of average fixation and extinction times of mutants in a general structured population of two types of species. The method is based on Markov chains and uses a mean field approximation to calculate the corresponding transition matrix. Analytical results are compared with the results of simulation of the Moran process on a number of population structures.Evolutionary Graph Theory (EGT) [1] is one of the most celebrated methods to study the evolution of species in structured populations. In this theory one considers a constant-size population of individuals which are connected to each other through a (directed) network which is called evolutionary graph [2]. A fitness is assigned to each type of species. At each time step one individual is selected for reproduction with a probability proportional to its fitness. Then it puts its offspring in place of one of its neighbors with a probability determined by evolutionary graph. This is in fact a generalization of the so called Moran Process [3] which takes place in a structured population instead of a well-mixed population.This theory has been vastly studied in recent years and different features and generalizations of it are addressed. Here we confine ourselves to populations constructed from two types of species whom we call residents and mutants. An interesting process is to start with just one mutant and see what the fate of the system is. In fact, the system will end up in one of the two possible states, namely, fixation or extinction of mutants. Two main quantities corresponding to this process are fixation probability and fixation time. Fixation probability is the probability for a single mutant to take over the whole population and fixation time is the average (conditional) time needed for this result. Both of these quantities are investigated by many researchers [4][5][6][7][8][9]. Obtaining fixation time is more challenging than fixation probability. In this letter we introduce a novel analytical method to find conditional fixation and extinction times and use it to obtain fixation time on a random graph in mean field approximation.Method: Consider a graph with N nodes. Each node can be one of two types, namely residents and mutants. Each type has its own fitness which is 1 for residents and r for mutants. A Moran process is running on top of this graph. At each time step one node is selected for reproduction with a probability proportional to its fitness. Then one of its neighbors is selected randomly and is replaced by the reproduced offspring. There are two important quantities corresponding to this process. The first one is fixation probability, i. e. the probability * mehdi.hajihashemi@ph.iut.ac.ir † samani@cc.iut.ac.ir

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“…Which is in agreement with the results obtained by other methods [4,8,11]. Now we examine this method on a random (Erdös-Rényi) network [13] and calculate the fixation time.…”

supporting

confidence: 86%

supporting

confidence: 55%

“…This can be easily seen for the present Markov chain using the recursive method of Ref. [8], namely by using the recursive relation…”

mentioning

confidence: 91%

mentioning

confidence: 99%

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Fixation time in evolutionary graphs: a mean field approach

jazi

Samani

2018

Preprint

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“…The transition probabilities of Y t are independent and identical [4,6]. Therefore Wald's 80 Eliminating time steps where X t = 0 does not affect the fixation probability of the Moran process.…”

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confidence: 99%

Wald’s martingale and the Moran process

Monk

Schaik

2020

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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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Exact numerical calculation of fixation probability and time on graphs

et al. 2016

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The Moran process on graphs is a popular model to study the dynamics of evolution in a spatially structured population. Exact analytical solutions for the fixation probability and time of a new mutant have been found for only a few classes of graphs so far. Simulations are time-expensive and many realizations are necessary, as the variance of the fixation times is high. We present an algorithm that numerically computes these quantities for arbitrary small graphs by an approach based on the transition matrix. The advantage over simulations is that the calculation has to be executed only once. Building the transition matrix is automated by our algorithm. This enables a fast and interactive study of different graph structures and their effect on fixation probability and time. We provide a fast implementation in C with this note [11]. Our code is very flexible, as it can handle two different update mechanisms (Birth-death or death-Birth), as well as arbitrary directed or undirected graphs.

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Analytical calculation of average fixation time in evolutionary graphs

Cited by 35 publications

References 12 publications

Fixation time in evolutionary graphs: a mean field approach

Fixation time in evolutionary graphs: a mean field approach

Wald’s martingale and the Moran process

Exact numerical calculation of fixation probability and time on graphs

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