# American Institute of Mathematical Sciences

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

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.
Citation: Fabio A. C. C. Chalub. An asymptotic expression for the fixation probability of a mutant in star graphs. Journal of Dynamics and 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), 113-151. doi: 10.4171/EMSS/3. [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), 1327-1346. doi: 10.1098/rspa.2009.0487. [3] M. Broom and J. Rychtář, Game-theoretical Models in Biology, CRC Press, Boca Raton, FL, 2013. [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), 2609-2627. doi: 10.1098/rspa.2008.0058. [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), 20130193, 11 pp. doi: 10.1098/rspa.2013.0193. [6] R. A. Fisher, The Genetical Theory of Natural Selection, Clarendon Press, Oxford, 1999. [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), 1762. doi: 10.1098/rspb.2013.0211. [8] B. Houchmandzadeh and M. Vallade, Exact results for fixation probability of bithermal evolutionary graphs, Biosystems, 112 (2013), 49-54. [9] E. Lieberman, C. Hauert and M. A. Nowak, Evolutionary dynamics on graphs, Nature, 433 (2005), 312-316. doi: 10.1038/nature03204. [10] P. A. P. Moran, The Statistical Processes of Evolutionary Theory, Clarendon, Oxford, 1962. [11] M. A. Nowak, Evolutionary Dynamics: Exploring the Equations of Life, The Belknap Press of Harvard University Press, Cambridge, MA, 2006. [12] P. Shakarian, P. Roos and A. Johnson, A review of evolutionary graph theory with applications to game theory, Biosystems, 107 (2012), 66-80. doi: 10.1016/j.biosystems.2011.09.006. [13] A. Traulsen, M. A. Nowak and J. M. Pacheco, Stochastic dynamics of invasion and fixation, Phys. Rev. E, 74 (2006), 011909. doi: 10.1103/PhysRevE.74.011909. [14] C. Zhang, Y. Wu, W. Liu and X. Yang, Fixation probabilities on complete star and bipartite digraphs, Discrete Dyn. Nat. Soc., (2012), Art. ID 940465, 21 pp. [15] S. Wright, Evolution in mendelian populations, Genetics, 16 (1931), 97-159.

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##### References:
 [1] B. Allen and M. Nowak, Games on graphs, EMS Surv. Math. Sci., 1 (2014), 113-151. doi: 10.4171/EMSS/3. [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), 1327-1346. doi: 10.1098/rspa.2009.0487. [3] M. Broom and J. Rychtář, Game-theoretical Models in Biology, CRC Press, Boca Raton, FL, 2013. [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), 2609-2627. doi: 10.1098/rspa.2008.0058. [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), 20130193, 11 pp. doi: 10.1098/rspa.2013.0193. [6] R. A. Fisher, The Genetical Theory of Natural Selection, Clarendon Press, Oxford, 1999. [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), 1762. doi: 10.1098/rspb.2013.0211. [8] B. Houchmandzadeh and M. Vallade, Exact results for fixation probability of bithermal evolutionary graphs, Biosystems, 112 (2013), 49-54. [9] E. Lieberman, C. Hauert and M. A. Nowak, Evolutionary dynamics on graphs, Nature, 433 (2005), 312-316. doi: 10.1038/nature03204. [10] P. A. P. Moran, The Statistical Processes of Evolutionary Theory, Clarendon, Oxford, 1962. [11] M. A. Nowak, Evolutionary Dynamics: Exploring the Equations of Life, The Belknap Press of Harvard University Press, Cambridge, MA, 2006. [12] P. Shakarian, P. Roos and A. Johnson, A review of evolutionary graph theory with applications to game theory, Biosystems, 107 (2012), 66-80. doi: 10.1016/j.biosystems.2011.09.006. [13] A. Traulsen, M. A. Nowak and J. M. Pacheco, Stochastic dynamics of invasion and fixation, Phys. Rev. E, 74 (2006), 011909. doi: 10.1103/PhysRevE.74.011909. [14] C. Zhang, Y. Wu, W. Liu and X. Yang, Fixation probabilities on complete star and bipartite digraphs, Discrete Dyn. Nat. Soc., (2012), Art. ID 940465, 21 pp. [15] S. Wright, Evolution in mendelian populations, Genetics, 16 (1931), 97-159.
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