Temporal and Dimensional Effects in Evolutionary Graph Theory

Evolutionary graph theory is the study of birth–death processes that are constrained by population structure. A principal problem in evolutionary graph theory is to obtain the probability that some initial population of mutants will fixate on a graph, and to determine how that fixation probability depends on the structure of that graph. A fluctuating mutant population on a graph can be considered as a random walk. Martingales exploit symmetry in the steps of a random walk to yield exact analytical expressions for fixation probabilities. They do not require simplifying assumptions such as large population sizes or weak selection. In this paper, we show how martingales can be used to obtain fixation probabilities for symmetric evolutionary graphs. We obtain simpler expressions for the fixation probabilities of star graphs and complete bipartite graphs than have been previously reported and show that these graphs do not amplify selection for advantageous mutations under all conditions.

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A review of evolutionary graph theory with applications to game theory

Biosystems ◽

10.1016/j.biosystems.2011.09.006 ◽

2012 ◽

Vol 107 (2) ◽

pp. 66-80 ◽

Cited By ~ 91

Author(s):

Paulo Shakarian ◽

Patrick Roos ◽

Anthony Johnson

Keyword(s):

Game Theory ◽

Graph Theory ◽

Evolutionary Graph Theory

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

Journal of Applied Mathematics ◽

10.1155/2014/901363 ◽

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.

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Evolutionary Graph Theory

SpringerBriefs in Computer Science - Diffusion in Social Networks ◽

10.1007/978-3-319-23105-1_6 ◽

2015 ◽

pp. 75-91

Author(s):

Paulo Shakarian ◽

Abhinav Bhatnagar ◽

Ashkan Aleali ◽

Elham Shaabani ◽

Ruocheng Guo

Keyword(s):

Graph Theory ◽

Evolutionary Graph Theory

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Environmental evolutionary graph theory

Journal of Theoretical Biology ◽

10.1016/j.jtbi.2014.06.040 ◽

2014 ◽

Vol 360 ◽

pp. 117-128 ◽

Cited By ~ 8

Author(s):

Wes Maciejewski ◽

Gregory J. Puleo

Keyword(s):

Graph Theory ◽

Evolutionary Graph Theory

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The two-mutant problem: clonal interference in evolutionary graph theory

Journal of Chemical Biology ◽

10.1007/s12154-010-0042-6 ◽

2010 ◽

Vol 3 (4) ◽

pp. 189-194 ◽

Cited By ~ 3

Author(s):

Chris Paley ◽

Sergei Taraskin ◽

Stephen Elliott

Keyword(s):

Graph Theory ◽

Clonal Interference ◽

Evolutionary Graph Theory

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FIXATION TIME FOR EVOLUTIONARY GRAPHS

International Journal of Modern Physics B ◽

10.1142/s0217979210056852 ◽

2010 ◽

Vol 24 (27) ◽

pp. 5285-5293 ◽

Cited By ~ 3

Author(s):

PU-YAN NIE ◽

PEI-AI ZHANG

Keyword(s):

Graph Theory ◽

Evolutionary Theory ◽

Biological Evolution ◽

Evolution Of Cooperation ◽

Fixation Time ◽

Social Phenomena ◽

Evolutionary Graph Theory

Evolutionary graph theory (EGT) is recently proposed by Lieberman et al. in 2005. EGT is successful for explaining biological evolution and some social phenomena. It is extremely important to consider the time of fixation for EGT in many practical problems, including evolutionary theory and the evolution of cooperation. This study characterizes the time to asymptotically reach fixation.

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