scholarly journals Evolutionary graph theory revisited: when is an evolutionary process equivalent to the Moran process?

2015 ◽  
Vol 471 (2182) ◽  
pp. 20150334 ◽  
Author(s):  
Karan Pattni ◽  
Mark Broom ◽  
Jan Rychtář ◽  
Lara J. Silvers
Keyword(s):  
Graph Theory ◽  
Evolutionary Dynamics ◽  
Evolutionary Process ◽  
Structured Populations ◽  
Fixation Probability ◽  
Finite Populations ◽  
Moran Model ◽  
Moran Process ◽  
Evolutionary Graph Theory ◽  
Associated Matrix

Evolution in finite populations is often modelled using the classical Moran process. Over the last 10 years, this methodology has been extended to structured populations using evolutionary graph theory. An important question in any such population is whether a rare mutant has a higher or lower chance of fixating (the fixation probability) than the Moran probability, i.e. that from the original Moran model, which represents an unstructured population. As evolutionary graph theory has developed, different ways of considering the interactions between individuals through a graph and an associated matrix of weights have been considered, as have a number of important dynamics. In this paper, we revisit the original paper on evolutionary graph theory in light of these extensions to consider these developments in an integrated way. In particular, we find general criteria for when an evolutionary graph with general weights satisfies the Moran probability for the set of six common evolutionary dynamics.

scholarly journals Fixation Probabilities of Evolutionary Graphs Based on the Positions of New Appearing Mutants

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Pei-ai Zhang
Keyword(s):  
Markov Chain ◽  
Graph Theory ◽  
Evolutionary Dynamics ◽  
Chain Process ◽  
Fixation Probability ◽  
Spatial Structures ◽  
Single Level ◽  
Fixation Probabilities ◽  
Evolutionary Graph Theory

Evolutionary graph theory is a nice measure to implement evolutionary dynamics on spatial structures of populations. To calculate the fixation probability is usually regarded as a Markov chain process, which is affected by the number of the individuals, the fitness of the mutant, the game strategy, and the structure of the population. However the position of the new mutant is important to its fixation probability. Here the position of the new mutant is laid emphasis on. The method is put forward to calculate the fixation probability of an evolutionary graph (EG) of single level. Then for a class of bilevel EGs, their fixation probabilities are calculated and some propositions are discussed. The conclusion is obtained showing that the bilevel EG is more stable than the corresponding one-rooted EG.

scholarly journals Evolutionary dynamics in structured populations

2010 ◽  
Vol 365 (1537) ◽  
pp. 19-30 ◽  
Author(s):  
Martin A. Nowak ◽  
Corina E. Tarnita ◽  
Tibor Antal
Keyword(s):  
Game Theory ◽  
Evolutionary Game Theory ◽  
Evolutionary Dynamics ◽  
Evolutionary Game ◽  
Evolutionary Process ◽  
Structured Populations ◽  
Evolution Of Cooperation ◽  
Evolutionary Graph Theory ◽  
Phenotype Space ◽  
Multicellular Organisms

Evolutionary dynamics shape the living world around us. At the centre of every evolutionary process is a population of reproducing individuals. The structure of that population affects evolutionary dynamics. The individuals can be molecules, cells, viruses, multicellular organisms or humans. Whenever the fitness of individuals depends on the relative abundance of phenotypes in the population, we are in the realm of evolutionary game theory. Evolutionary game theory is a general approach that can describe the competition of species in an ecosystem, the interaction between hosts and parasites, between viruses and cells, and also the spread of ideas and behaviours in the human population. In this perspective, we review the recent advances in evolutionary game dynamics with a particular emphasis on stochastic approaches in finite sized and structured populations. We give simple, fundamental laws that determine how natural selection chooses between competing strategies. We study the well-mixed population, evolutionary graph theory, games in phenotype space and evolutionary set theory. We apply these results to the evolution of cooperation. The mechanism that leads to the evolution of cooperation in these settings could be called ‘spatial selection’: cooperators prevail against defectors by clustering in physical or other spaces.

scholarly journals The effect of population structure on the rate of evolution

2013 ◽  
Vol 280 (1762) ◽  
pp. 20130211 ◽  
Author(s):  
Marcus Frean ◽  
Paul B. Rainey ◽  
Arne Traulsen
Keyword(s):  
Population Structure ◽  
Evolutionary Process ◽  
Ecological Factors ◽  
Structured Populations ◽  
Fixation Time ◽  
Fixation Probability ◽  
Star Graphs ◽  
Rate Of Evolution ◽  
Population Structures ◽  
Evolution Proceeds

Ecological factors exert a range of effects on the dynamics of the evolutionary process. A particularly marked effect comes from population structure, which can affect the probability that new mutations reach fixation. Our interest is in population structures, such as those depicted by ‘star graphs’, that amplify the effects of selection by further increasing the fixation probability of advantageous mutants and decreasing the fixation probability of disadvantageous mutants. The fact that star graphs increase the fixation probability of beneficial mutations has lead to the conclusion that evolution proceeds more rapidly in star-structured populations, compared with mixed (unstructured) populations. Here, we show that the effects of population structure on the rate of evolution are more complex and subtle than previously recognized and draw attention to the importance of fixation time. By comparing population structures that amplify selection with other population structures, both analytically and numerically, we show that evolution can slow down substantially even in populations where selection is amplified.

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

2018 ◽  
Author(s):  
David Hathcock ◽  
Steven H. Strogatz
Keyword(s):  
Complete Graph ◽  
Time Distribution ◽  
Evolutionary Dynamics ◽  
Structured Populations ◽  
Relative Fitness ◽  
Death Process ◽  
Fixation Time ◽  
Large Network ◽  
Moran Process ◽  
Fitness Dependence

Evolutionary graph theory models the effects of natural selection and random drift on structured populations of mutant and non-mutant individuals. Recent studies have shown that fixation times, which determine the rate of evolution, 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, each of which admits an exact solution 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 weighted convolution of two Gumbel distributions, with a weight depending on the relative fitness. When fitness is neutral, however, the Moran process has a highly skewed fixation-time distribution on both the complete graph and the ring. In this sense, the case of neutral fitness is singular. Even on these simple network structures, the fixation-time distribution exhibits rich fitness dependence, with discontinuities and regions of universality. Applications of our methods to a multi-fitness Moran model, times to partial fixation, and evolution on random networks are discussed.

scholarly journals Structural symmetry in evolutionary games

2015 ◽  
Vol 12 (111) ◽  
pp. 20150420 ◽  
Author(s):  
Alex McAvoy ◽  
Christoph Hauert
Keyword(s):  
Population Structure ◽  
Evolutionary Game ◽  
Evolutionary Process ◽  
Evolutionary Games ◽  
Structured Populations ◽  
Fixation Probability ◽  
Matrix Games ◽  
Single Mutant ◽  
Symmetric Games ◽  
Monomorphic Population

In evolutionary game theory, an important measure of a mutant trait (strategy) is its ability to invade and take over an otherwise-monomorphic population. Typically, one quantifies the success of a mutant strategy via the probability that a randomly occurring mutant will fixate in the population. However, in a structured population, this fixation probability may depend on where the mutant arises. Moreover, the fixation probability is just one quantity by which one can measure the success of a mutant; fixation time , for instance, is another. We define a notion of homogeneity for evolutionary games that captures what it means for two single-mutant states, i.e. two configurations of a single mutant in an otherwise-monomorphic population, to be ‘evolutionarily equivalent’ in the sense that all measures of evolutionary success are the same for both configurations. Using asymmetric games, we argue that the term ‘homogeneous’ should apply to the evolutionary process as a whole rather than to just the population structure. For evolutionary matrix games in graph-structured populations, we give precise conditions under which the resulting process is homogeneous. Finally, we show that asymmetric matrix games can be reduced to symmetric games if the population structure possesses a sufficient degree of symmetry.

scholarly journals On the fixation probability of superstars

2013 ◽  
Vol 469 (2156) ◽  
pp. 20130193 ◽  
Author(s):  
Josep Díaz ◽  
Leslie Ann Goldberg ◽  
George B. Mertzios ◽  
David Richerby ◽  
Maria Serna ◽  
...  
Keyword(s):  
Computer Algebra ◽  
Evolutionary Dynamics ◽  
Process Models ◽  
Relative Fitness ◽  
Genetic Mutations ◽  
Fixation Probability ◽  
Mutant Population ◽  
Moran Process ◽  
Limiting Behaviour ◽  
Symbolic Matrix

The Moran process models the spread of genetic mutations through populations. A mutant with relative fitness r is introduced and the system evolves, either reaching fixation (an all-mutant population) or extinction (no mutants). In a widely cited paper, Lieberman et al. (2005 Evolutionary dynamics on graphs. Nature 433 , 312–316) generalize the model to populations on the vertices of graphs. They describe a class of graphs (‘superstars’), with a parameter k and state that the fixation probability tends to 1− r − k as the graphs get larger: we show that this is untrue as stated. Specifically, for k =5, we show that the fixation probability (in the limit, as graphs get larger) cannot exceed 1−1/ j ( r ), where j ( r )= Θ ( r 4 ), contrary to the claimed result. Our proof is fully rigorous, though we use a computer algebra package to invert a 31×31 symbolic matrix. We do believe the qualitative claim of Lieberman et al. —that superstar fixation probability tends to 1 as k increases—and that it can probably be proved similarly to their sketch. We were able to run larger simulations than the ones they presented. Simulations on graphs of around 40 000 vertices do not support their claim but these graphs might be too small to exhibit the limiting behaviour.

scholarly journals The Moran process on 2-chromatic graphs

2020 ◽  
Vol 16 (11) ◽  
pp. e1008402
Author(s):  
Kamran Kaveh ◽  
Alex McAvoy ◽  
Krishnendu Chatterjee ◽  
Martin A. Nowak
Keyword(s):  
Graph Coloring ◽  
Evolutionary Dynamics ◽  
Population Heterogeneity ◽  
Graph Connectivity ◽  
Regular Graphs ◽  
Breeding Sites ◽  
Moran Model ◽  
Colored Graphs ◽  
Moran Process

Resources are rarely distributed uniformly within a population. Heterogeneity in the concentration of a drug, the quality of breeding sites, or wealth can all affect evolutionary dynamics. In this study, we represent a collection of properties affecting the fitness at a given location using a color. A green node is rich in resources while a red node is poorer. More colors can represent a broader spectrum of resource qualities. For a population evolving according to the birth-death Moran model, the first question we address is which structures, identified by graph connectivity and graph coloring, are evolutionarily equivalent. We prove that all properly two-colored, undirected, regular graphs are evolutionarily equivalent (where “properly colored” means that no two neighbors have the same color). We then compare the effects of background heterogeneity on properly two-colored graphs to those with alternative schemes in which the colors are permuted. Finally, we discuss dynamic coloring as a model for spatiotemporal resource fluctuations, and we illustrate that random dynamic colorings often diminish the effects of background heterogeneity relative to a proper two-coloring.

scholarly journals Fixation probabilities in network structured meta-populations

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Sedigheh Yagoobi ◽  
Arne Traulsen
Keyword(s):  
Evolutionary Dynamics ◽  
Evolutionary Ecology ◽  
Research Topic ◽  
Fixation Probability ◽  
Popular Approach ◽  
Fixation Probabilities ◽  
Evolutionary Graph Theory ◽  
Selection For ◽  
Carry Over ◽  
Regular Networks

AbstractThe effect of population structure on evolutionary dynamics is a long-lasting research topic in evolutionary ecology and population genetics. Evolutionary graph theory is a popular approach to this problem, where individuals are located on the nodes of a network and can replace each other via the links. We study the effect of complex network structure on the fixation probability, but instead of networks of individuals, we model a network of sub-populations with a probability of migration between them. We ask how the structure of such a meta-population and the rate of migration affect the fixation probability. Many of the known results for networks of individuals carry over to meta-populations, in particular for regular networks or low symmetric migration probabilities. However, when patch sizes differ we find interesting deviations between structured meta-populations and networks of individuals. For example, a two patch structure with unequal population size suppresses selection for low migration probabilities.

scholarly journals Counterintuitive properties of the fixation time in network-structured populations

2014 ◽  
Vol 11 (99) ◽  
pp. 20140606 ◽  
Author(s):  
Laura Hindersin ◽  
Arne Traulsen
Keyword(s):  
Evolutionary Dynamics ◽  
Transition Probabilities ◽  
Structured Populations ◽  
Death Process ◽  
Fixation Time ◽  
Fixation Probability ◽  
Mutant Type ◽  
Starting Conditions ◽  
Fixation Probabilities ◽  
Regular Networks

Evolutionary dynamics on graphs can lead to many interesting and counterintuitive findings. We study the Moran process, a discrete time birth–death process, that describes the invasion of a mutant type into a population of wild-type individuals. Remarkably, the fixation probability of a single mutant is the same on all regular networks. But non-regular networks can increase or decrease the fixation probability. While the time until fixation formally depends on the same transition probabilities as the fixation probabilities, there is no obvious relation between them. For example, an amplifier of selection, which increases the fixation probability and thus decreases the number of mutations needed until one of them is successful, can at the same time slow down the process of fixation. Based on small networks, we show analytically that (i) the time to fixation can decrease when links are removed from the network and (ii) the node providing the best starting conditions in terms of the shortest fixation time depends on the fitness of the mutant. Our results are obtained analytically on small networks, but numerical simulations show that they are qualitatively valid even in much larger populations.

EVOLUTIONARY GRAPHS WITH FREQUENCY DEPENDENT FITNESS

2009 ◽  
Vol 23 (04) ◽  
pp. 537-543 ◽  
Author(s):  
PU-YAN NIE ◽  
PEI-AI ZHANG
Keyword(s):  
Graph Theory ◽  
Large Population ◽  
Fixation Probability ◽  
Frequency Dependent ◽  
Evolutionary Graph Theory

Evolutionary graph theory was recently proposed by Lieberman et al. in 2005. In the previous papers about evolutionary graphs (EGs), the fitness of the residents in the EGs is in general assumed to be unity, and the fitness of a mutant is assumed to be a constant r. We aim to extend EG to general cases in this paper, namely, the fitness of a mutant is heavily dependent upon frequency. The corresponding properties for these new EGs are analyzed, and the fixation probability is obtained for large population.

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