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Approximating Fixation Probabilities in the Generalized Moran Process

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  • First Online:
Encyclopedia of Algorithms

  • 159 Accesses

Years and Authors of Summarized Original Work

  • 2014; Diaz, Goldberg, Mertzios, Richerby, Serna, Spirakis

Problem Definition

Population and evolutionary dynamics have been extensively studied, usually with the assumption that the evolving population has no spatial structure. One of the main models in this area is the Moran process [17]. The initial population contains a single “mutant” with fitness r > 0, with all other individuals having fitness 1. At each step of the process, an individual is chosen at random, with probability proportional to its fitness. This individual reproduces, replacing a second individual, chosen uniformly at random, with a copy of itself.

Lieberman, Hauert, and Nowak introduced a generalization of the Moran process, where the members of the population are placed on the vertices of a connected graph which is, in general, directed [13, 19]. In this model, the initial population again consists of a single mutant of fitness r> 0 placed on a vertex chosen...

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Recommended Reading

  1. Aldous DJ, Fill JA (2002) Reversible Markov chains and random walks on graphs. Monograph in preparation. Available at http://www.stat.berkeley.edu/aldous/RWG/book.html

    Google Scholar 

  2. Berger E (2001) Dynamic monopolies of constant size. J Comb Theory Ser B 83:191–200

    Article  MathSciNet  MATH  Google Scholar 

  3. Broom M, Hadjichrysanthou C, Rychtář J (2010) Evolutionary games on graphs and the speed of the evolutionary process. Proc R Soc A 466(2117):1327–1346

    Article  MathSciNet  MATH  Google Scholar 

  4. Broom M, Hadjichrysanthou C, Rychtář J (2010) Two results on evolutionary processes on general non-directed graphs. Proc R Soc A 466(2121):2795–2798

    Article  MATH  Google Scholar 

  5. Broom M, Rychtář J (2008) An analysis of the fixation probability of a mutant on special classes of non-directed graphs. Proc R Soc A 464(2098):2609–2627

    Article  MathSciNet  MATH  Google Scholar 

  6. Broom M, Rychtář J, Stadler B (2009) Evolutionary dynamics on small order graphs. J Interdiscip Math 12:129–140

    Article  MATH  Google Scholar 

  7. Durrett R (1988) Lecture notes on particle systems and percolation. Wadsworth Publishing Company, Pacific Grove

    MATH  Google Scholar 

  8. Easley D, Kleinberg J (2010) Networks, crowds, and markets: reasoning about a highly connected world. Cambridge University Press, New York

    Book  MATH  Google Scholar 

  9. Hajek B (1982) Hitting-time and occupation-time bounds implied by drift analysis with applications. Adv Appl Probab 14(3):502–525

    Article  MathSciNet  MATH  Google Scholar 

  10. He J, Yao X (2001) Drift analysis and average time complexity of evolutionary algorithms. Artif Intell 127:57–85

    Article  MathSciNet  MATH  Google Scholar 

  11. Karp RM, Luby M (1983) Monte-Carlo algorithms for enumeration and reliability problems. In: Proceedings of 24th annual IEEE symposium on foundations of computer science (FOCS), Tucson, pp 56–64

    Google Scholar 

  12. Kempel D, Kleinberg J, Tardos E (2005) Influential nodes in a diffusion model for social networks. In: Proceedings of the 32nd international colloquium on automata, languages and programming (ICALP), Lisbon. Lecture notes in computer science, vol 3580, pp 1127–1138. Springer

    Google Scholar 

  13. Lieberman E, Hauert C, Nowak MA (2005) Evolutionary dynamics on graphs. Nature 433:312–316

    Article  Google Scholar 

  14. Liggett TM (1985) Interacting particle systems. Springer, New York

    Book  MATH  Google Scholar 

  15. Luby M, Randall D, Sinclair A (2001) Markov chain algorithms for planar lattice structures. SIAM J Comput 31(1):167–192

    Article  MathSciNet  MATH  Google Scholar 

  16. Mertzios GB, Nikoletseas S, Raptopoulos C, Spirakis PG (2013) Natural models for evolution on networks. Theor Comput Sci 477:76–95

    Article  MathSciNet  MATH  Google Scholar 

  17. Moran PAP (1958) Random processes in genetics. Proc Camb Philos Soc 54(1):60–71

    Article  MathSciNet  MATH  Google Scholar 

  18. Mossel E, Roch S (2007) On the submodularity of influence in social networks. In: Proceedings of the 39th annual ACM symposium on theory of computing (STOC), San Diego, pp 128–134

    Google Scholar 

  19. Nowak MA (2006) Evolutionary dynamics: exploring the equations of life. Harvard University Press, Cambridge

    MATH  Google Scholar 

  20. Taylor C, Iwasa Y, Nowak MA (2006) A symmetry of fixation times in evolutionary dynamics. J Theor Biol 243(2):245–251

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Engineering and Computing Sciences, Durham University, Science Site, South Road, Durham, UK

    GeorgeB. Mertzios

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  1. GeorgeB. Mertzios
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Correspondence to GeorgeB. Mertzios .

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Editors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, IL, USA

    Ming-Yang Kao

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Mertzios, G. (2016). Approximating Fixation Probabilities in the Generalized Moran Process. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_596

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