American Institute of Mathematical Sciences

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

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

Fabio A. C. C. Chalub 1,

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.
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

[1]

Deena Schmidt, Janet Best, Mark S. Blumberg. Random graph and stochastic process contributions to network dynamics. Conference Publications, 2011, 2011 (Special) : 1279-1288. doi: 10.3934/proc.2011.2011.1279
[2]

Jaime Angulo Pava, Nataliia Goloshchapova. On the orbital instability of excited states for the NLS equation with the δ-interaction on a star graph. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5039-5066. doi: 10.3934/dcds.2018221
[3]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020033
[4]

Amin Boumenir, Vu Kim Tuan. Reconstruction of the coefficients of a star graph from observations of its vertices. Inverse Problems & Imaging, 2018, 12 (6) : 1293-1308. doi: 10.3934/ipi.2018054
[5]

Mirela Domijan, Markus Kirkilionis. Graph theory and qualitative analysis of reaction networks. Networks & Heterogeneous Media, 2008, 3 (2) : 295-322. doi: 10.3934/nhm.2008.3.295
[6]

M. D. König, Stefano Battiston, M. Napoletano, F. Schweitzer. On algebraic graph theory and the dynamics of innovation networks. Networks & Heterogeneous Media, 2008, 3 (2) : 201-219. doi: 10.3934/nhm.2008.3.201
[7]

Barton E. Lee. Consensus and voting on large graphs: An application of graph limit theory. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1719-1744. doi: 10.3934/dcds.2018071
[8]

Giuseppe Maria Coclite, Carlotta Donadello. Vanishing viscosity on a star-shaped graph under general transmission conditions at the node. Networks & Heterogeneous Media, 2020, 15 (2) : 197-213. doi: 10.3934/nhm.2020009
[9]

Lu Yang, Guangsheng Wei, Vyacheslav Pivovarchik. Direct and inverse spectral problems for a star graph of Stieltjes strings damped at a pendant vertex. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020063
[10]

Eric Babson and Dmitry N. Kozlov. Topological obstructions to graph colorings. Electronic Research Announcements, 2003, 9: 61-68.

[11]

J. William Hoffman. Remarks on the zeta function of a graph. Conference Publications, 2003, 2003 (Special) : 413-422. doi: 10.3934/proc.2003.2003.413
[12]

Oded Schramm. Hyperfinite graph limits. Electronic Research Announcements, 2008, 15: 17-23. doi: 10.3934/era.2008.15.17
[13]

John Kieffer and En-hui Yang. Ergodic behavior of graph entropy. Electronic Research Announcements, 1997, 3: 11-16.

[14]

Jan Rychtář, Dewey T. Taylor. Moran process and Wright-Fisher process favor low variability. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020242
[15]

A. C. Eberhard, J-P. Crouzeix. Existence of closed graph, maximal, cyclic pseudo-monotone relations and revealed preference theory. Journal of Industrial & Management Optimization, 2007, 3 (2) : 233-255. doi: 10.3934/jimo.2007.3.233
[16]

Roy H. Goodman. NLS bifurcations on the bowtie combinatorial graph and the dumbbell metric graph. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2203-2232. doi: 10.3934/dcds.2019093
[17]

Mario Roy, Mariusz Urbański. Random graph directed Markov systems. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 261-298. doi: 10.3934/dcds.2011.30.261
[18]

Dominique Zosso, Braxton Osting. A minimal surface criterion for graph partitioning. Inverse Problems & Imaging, 2016, 10 (4) : 1149-1180. doi: 10.3934/ipi.2016036
[19]

Mario Jorge Dias Carneiro, Rafael O. Ruggiero. On the graph theorem for Lagrangian minimizing tori. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6029-6045. doi: 10.3934/dcds.2018260
[20]

Chun-Xiang Guo, Guo Qiang, Jin Mao-Zhu, Zhihan Lv. Dynamic systems based on preference graph and distance. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1139-1154. doi: 10.3934/dcdss.2015.8.1139

 Impact Factor: 

Tools

Metrics

  • PDF downloads (37)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences