Mutants and Residents with Different Connection Graphs in the Moran Process

  • Themistoklis Melissourgos
  • Sotiris Nikoletseas
  • Christoforos Raptopoulos
  • Paul Spirakis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)

Abstract

The Moran process, as studied by Lieberman et al. [10], is a stochastic process modeling the spread of genetic mutations in populations. In this process, agents of a two-type population (i.e. mutants and residents) are associated with the vertices of a graph. Initially, only one vertex chosen uniformly at random (u.a.r.) is a mutant, with fitness \(r > 0\), while all other individuals are residents, with fitness 1. In every step, an individual is chosen with probability proportional to its fitness, and its state (mutant or resident) is passed on to a neighbor which is chosen u.a.r. In this paper, we introduce and study for the first time a generalization of the model of [10] by assuming that different types of individuals perceive the population through different graphs defined on the same vertex set, namely \(G_R = (V, E_R)\) for residents and \(G_M = (V, E_M)\) for mutants. In this model, we study the fixation probability, namely the probability that eventually only mutants remain in the population, for various pairs of graphs.

In particular, in the first part of the paper, we examine how known results from the original single-graph model of [10] can be transferred to our 2-graph model. In that direction, by using a Markov chain abstraction, we provide a generalization of the Isothermal Theorem of [10], that gives sufficient conditions for a pair of graphs to have fixation probability equal to the fixation probability of a pair of cliques; this corresponds to the absorption probability of a birth-death process with forward bias r.

In the second part of the paper, we give a 2-player strategic game view of the process where player payoffs correspond to fixation and/or extinction probabilities. In this setting, we attempt to identify best responses for each player. We give evidence that the clique is the most beneficial graph for both players, by proving bounds on the fixation probability when one of the two graphs is complete and the other graph belongs to various natural graph classes.

In the final part of the paper, we examine the possibility of efficient approximation of the fixation probability. Interestingly, we show that there is a pair of graphs for which the fixation probability is exponentially small. This implies that the fixation probability in the general case of an arbitrary pair of graphs cannot be approximated via a method similar to [2]. Nevertheless, we prove that, in the special case when the mutant graph is complete, an efficient approximation of the fixation probability is possible through an FPRAS which we describe.

Keywords

Moran process Fixation probability Evolutionary dynamics 

References

  1. 1.
    Broom, M., Rychtář, J.: An analysis of the fixation probability of a mutant on special classes of non-directed graphs. Proc. Roy. Soc. Lond. A: Math. Phys. Eng. Sci. 464(2098), 2609–2627 (2008)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Díaz, J., Goldberg, L.A., Mertzios, G.B., Richerby, D., Serna, M., Spirakis, P.G.: Approximating fixation probabilities in the generalized Moran process. Algorithmica 69(1), 78–91 (2014)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Easley, D., Kleinberg, J.: Networks, Crowds, and Markets, vol. 6, no. 1, pp. 1–6. Cambridge University Press, Cambridge (2010)Google Scholar
  4. 4.
    Galanis, A., Göbel, A., Goldberg, L.A., Lapinskas, J., Richerby, D.: Amplifiers for the Moran process. In: 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, 11–15 July 2016, Rome, Italy, pp. 62:1–62:13 (2016)Google Scholar
  5. 5.
    Giakkoupis, G.: Amplifiers and suppressors of selection for the Moran process on undirected graphs. CoRR, abs/1611.01585 (2016)Google Scholar
  6. 6.
    Goldberg, L.A., Lapinskas, J., Lengler, J., Meier, F., Panagiotou, K., Pfister, P.: Asymptotically optimal amplifiers for the Moran process. ArXiv e-prints, November 2016Google Scholar
  7. 7.
    Hofbauer, J., Sigmund, K.: Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge (1998)CrossRefMATHGoogle Scholar
  8. 8.
    Ibsen-Jensen, R., Chatterjee, K., Nowak, M.A.: Computational complexity of ecological and evolutionary spatial dynamics. Proc. Nat. Acad. Sci. 112(51), 15636–15641 (2015)Google Scholar
  9. 9.
    Karp, R.M., Luby, M.: Monte-Carlo algorithms for enumeration and reliability problems. In: 24th Annual Symposium on Foundations of Computer Science, pp. 56–64. IEEE (1983)Google Scholar
  10. 10.
    Lieberman, E., Hauert, C., Nowak, M.A.: Evolutionary dynamics on graphs. Nature 433(7023), 312–316 (2005)CrossRefGoogle Scholar
  11. 11.
    Melissourgos, T., Nikoletseas, S.E., Raptopoulos, C., Spirakis, P.G.: Mutants and residents with different connection graphs in the Moran process. CoRR, abs/1710.07365 (2017)Google Scholar
  12. 12.
    Mertzios, G.B., Nikoletseas, S., Raptopoulos, C., Spirakis, P.G.: Natural models for evolution on networks. In: Chen, N., Elkind, E., Koutsoupias, E. (eds.) WINE 2011. LNCS, vol. 7090, pp. 290–301. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-25510-6_25 CrossRefGoogle Scholar
  13. 13.
    Mertzios, G.B., Spirakis, P.G.: Strong bounds for evolution in networks. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013. LNCS, vol. 7966, pp. 669–680. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-39212-2_58 CrossRefGoogle Scholar
  14. 14.
    Moran, P.A.P.: Random processes in genetics. Math. Proc. Camb. Philos. Soc. 54(1), 6071 (1958)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Nowak, M.A.: Evolutionary Dynamics: Exploring the Equations of Life. Harvard University Press, Cambridge (2006)MATHGoogle Scholar
  16. 16.
    Ohtsuki, H., Pacheco, J.M., Nowak, M.A.: Evolutionary graph theory: breaking the symmetry between interaction and replacement. J. Theor. Biol. 246(4), 681–694 (2007)MathSciNetCrossRefGoogle Scholar

Copyright information

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Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of LiverpoolLiverpoolUK
  2. 2.Computer Technology Institute and Press “Diophantus” (CTI)PatrasGreece
  3. 3.Computer Engineering and Informatics DepartmentUniversity of PatrasPatrasGreece

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