An asymptotic expression for the fixation probability of a mutant in star graphs

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July 2016, 3(3): 217-223. doi: 10.3934/jdg.2016011

An asymptotic expression for the fixation probability of a mutant in star graphs

1.	Departamento de Matemática and Centro de Matemática e Aplicações, Universidade Nova de Lisboa, Quinta da Torre, 2829-516, Caparica, Portugal

Received July 2015 Revised February 2016 Published July 2016

Abstract
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We consider the Moran process in a graph called the ``star'' and obtain the asymptotic expression for the fixation probability of a single mutant when the size of the graph is large. The expression obtained corrects the previously known expression announced in reference [E Lieberman, C Hauert, and MA Nowak. Evolutionary dynamics on graphs. Nature, 433(7023):312–316, 2005] and further studied in [M. Broom and J. Rychtar. An analysis of the fixation probability of a mutant on special classes of non-directed graphs. Proc. R. Soc. A-Math. Phys. Eng. Sci., 464(2098):2609–2627, 2008]. We also show that the star graph is an accelerator of evolution, if the graph is large enough.

Keywords: Evolutionary graph theory, asymptotic expansions, fixation probability, Moran process, star graph..

Mathematics Subject Classification: 92D25, 41A6.

Citation: Fabio A. C. C. Chalub. An asymptotic expression for the fixation probability of a mutant in star graphs. Journal of Dynamics & Games, 2016, 3 (3) : 217-223. doi: 10.3934/jdg.2016011

References:

[1]	B. Allen and M. Nowak, Games on graphs,, EMS Surv. Math. Sci., 1 (2014), 113. doi: 10.4171/EMSS/3. Google Scholar
[2]	M. Broom, C. Hadjichrysanthou and J. Rychtar, Evolutionary games on graphs and the speed of the evolutionary process,, Proc. R. Soc. A-Math. Phys. Eng. Sci., 466 (2010), 1327. doi: 10.1098/rspa.2009.0487. Google Scholar
[3]	M. Broom and J. Rychtář, Game-theoretical Models in Biology,, CRC Press, (2013). Google Scholar
[4]	M. Broom and J. Rychtar, An analysis of the fixation probability of a mutant on special classes of non-directed graphs,, Proc. R. Soc. A-Math. Phys. Eng. Sci., 464 (2008), 2609. doi: 10.1098/rspa.2008.0058. Google Scholar
[5]	J. Diaz, L. A. Goldberg, G. B. Mertzios, D. Richerby, M. Serna and P. G. Spirakis, On the fixation probability of superstars,, Proc. R. Soc. A-Math. Phys. Eng. Sci., 469 (2013). doi: 10.1098/rspa.2013.0193. Google Scholar
[6]	R. A. Fisher, The Genetical Theory of Natural Selection,, Clarendon Press, (1999). Google Scholar
[7]	M. Frean, P. B. Rainey and A. Traulsen, The effect of population structure on the rate of evolution,, Proc. R. Soc. B-Biol. Sci., 280 (2013). doi: 10.1098/rspb.2013.0211. Google Scholar
[8]	B. Houchmandzadeh and M. Vallade, Exact results for fixation probability of bithermal evolutionary graphs,, Biosystems, 112 (2013), 49. Google Scholar
[9]	E. Lieberman, C. Hauert and M. A. Nowak, Evolutionary dynamics on graphs,, Nature, 433 (2005), 312. doi: 10.1038/nature03204. Google Scholar
[10]	P. A. P. Moran, The Statistical Processes of Evolutionary Theory,, Clarendon, (1962). Google Scholar
[11]	M. A. Nowak, Evolutionary Dynamics: Exploring the Equations of Life,, The Belknap Press of Harvard University Press, (2006). Google Scholar
[12]	P. Shakarian, P. Roos and A. Johnson, A review of evolutionary graph theory with applications to game theory,, Biosystems, 107 (2012), 66. doi: 10.1016/j.biosystems.2011.09.006. Google Scholar
[13]	A. Traulsen, M. A. Nowak and J. M. Pacheco, Stochastic dynamics of invasion and fixation,, Phys. Rev. E, 74 (2006). doi: 10.1103/PhysRevE.74.011909. Google Scholar
[14]	C. Zhang, Y. Wu, W. Liu and X. Yang, Fixation probabilities on complete star and bipartite digraphs,, Discrete Dyn. Nat. Soc., (2012). Google Scholar
[15]	S. Wright, Evolution in mendelian populations,, Genetics, 16 (1931), 97. Google Scholar

show all references

References:

[1]	B. Allen and M. Nowak, Games on graphs,, EMS Surv. Math. Sci., 1 (2014), 113. doi: 10.4171/EMSS/3. Google Scholar
[2]	M. Broom, C. Hadjichrysanthou and J. Rychtar, Evolutionary games on graphs and the speed of the evolutionary process,, Proc. R. Soc. A-Math. Phys. Eng. Sci., 466 (2010), 1327. doi: 10.1098/rspa.2009.0487. Google Scholar
[3]	M. Broom and J. Rychtář, Game-theoretical Models in Biology,, CRC Press, (2013). Google Scholar
[4]	M. Broom and J. Rychtar, An analysis of the fixation probability of a mutant on special classes of non-directed graphs,, Proc. R. Soc. A-Math. Phys. Eng. Sci., 464 (2008), 2609. doi: 10.1098/rspa.2008.0058. Google Scholar
[5]	J. Diaz, L. A. Goldberg, G. B. Mertzios, D. Richerby, M. Serna and P. G. Spirakis, On the fixation probability of superstars,, Proc. R. Soc. A-Math. Phys. Eng. Sci., 469 (2013). doi: 10.1098/rspa.2013.0193. Google Scholar
[6]	R. A. Fisher, The Genetical Theory of Natural Selection,, Clarendon Press, (1999). Google Scholar
[7]	M. Frean, P. B. Rainey and A. Traulsen, The effect of population structure on the rate of evolution,, Proc. R. Soc. B-Biol. Sci., 280 (2013). doi: 10.1098/rspb.2013.0211. Google Scholar
[8]	B. Houchmandzadeh and M. Vallade, Exact results for fixation probability of bithermal evolutionary graphs,, Biosystems, 112 (2013), 49. Google Scholar
[9]	E. Lieberman, C. Hauert and M. A. Nowak, Evolutionary dynamics on graphs,, Nature, 433 (2005), 312. doi: 10.1038/nature03204. Google Scholar
[10]	P. A. P. Moran, The Statistical Processes of Evolutionary Theory,, Clarendon, (1962). Google Scholar
[11]	M. A. Nowak, Evolutionary Dynamics: Exploring the Equations of Life,, The Belknap Press of Harvard University Press, (2006). Google Scholar
[12]	P. Shakarian, P. Roos and A. Johnson, A review of evolutionary graph theory with applications to game theory,, Biosystems, 107 (2012), 66. doi: 10.1016/j.biosystems.2011.09.006. Google Scholar
[13]	A. Traulsen, M. A. Nowak and J. M. Pacheco, Stochastic dynamics of invasion and fixation,, Phys. Rev. E, 74 (2006). doi: 10.1103/PhysRevE.74.011909. Google Scholar
[14]	C. Zhang, Y. Wu, W. Liu and X. Yang, Fixation probabilities on complete star and bipartite digraphs,, Discrete Dyn. Nat. Soc., (2012). Google Scholar
[15]	S. Wright, Evolution in mendelian populations,, Genetics, 16 (1931), 97. Google Scholar

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