Encyclopedia of Algorithms

Living Edition
| Editors: Ming-Yang Kao

Approximating Fixation Probabilities in the Generalized Moran Process

  • George B. Mertzios
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27848-8_596-1

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...

Keywords

Evolutionary dynamics Moran process Fixation probability Markov-chain Monte Carlo Approximation algorithm 
This is a preview of subscription content, log in to check access.

Recommended Reading

  1. 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
  2. 2.
    Berger E (2001) Dynamic monopolies of constant size. J Comb Theory Ser B 83:191–200CrossRefzbMATHGoogle Scholar
  3. 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):1327–1346CrossRefzbMATHGoogle Scholar
  4. 4.
    Broom M, Hadjichrysanthou C, Rychtář J (2010) Two results on evolutionary processes on general non-directed graphs. Proc R Soc A 466(2121):2795–2798CrossRefzbMATHGoogle Scholar
  5. 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):2609–2627CrossRefMathSciNetGoogle Scholar
  6. 6.
    Broom M, Rychtář J, Stadler B (2009) Evolutionary dynamics on small order graphs. J Interdiscip Math 12:129–140CrossRefzbMATHGoogle Scholar
  7. 7.
    Durrett R (1988) Lecture notes on particle systems and percolation. Wadsworth Publishing Company, Pacific GrovezbMATHGoogle Scholar
  8. 8.
    Easley D, Kleinberg J (2010) Networks, crowds, and markets: reasoning about a highly connected world. Cambridge University Press, New YorkCrossRefGoogle Scholar
  9. 9.
    Hajek B (1982) Hitting-time and occupation-time bounds implied by drift analysis with applications. Adv Appl Probab 14(3):502–525CrossRefzbMATHGoogle Scholar
  10. 10.
    He J, Yao X (2001) Drift analysis and average time complexity of evolutionary algorithms. Artif Intell 127:57–85CrossRefzbMATHMathSciNetGoogle Scholar
  11. 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 56–64Google Scholar
  12. 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. SpringerGoogle Scholar
  13. 13.
    Lieberman E, Hauert C, Nowak MA (2005) Evolutionary dynamics on graphs. Nature 433:312–316CrossRefGoogle Scholar
  14. 14.
    Liggett TM (1985) Interacting particle systems. Springer, New YorkCrossRefzbMATHGoogle Scholar
  15. 15.
    Luby M, Randall D, Sinclair A (2001) Markov chain algorithms for planar lattice structures. SIAM J Comput 31(1):167–192CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Mertzios GB, Nikoletseas S, Raptopoulos C, Spirakis PG (2013) Natural models for evolution on networks. Theor Comput Sci 477:76–95CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Moran PAP (1958) Random processes in genetics. Proc Camb Philos Soc 54(1):60–71CrossRefzbMATHGoogle Scholar
  18. 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 128–134Google Scholar
  19. 19.
    Nowak MA (2006) Evolutionary dynamics: exploring the equations of life. Harvard University Press, CambridgeGoogle Scholar
  20. 20.
    Taylor C, Iwasa Y, Nowak MA (2006) A symmetry of fixation times in evolutionary dynamics. J Theor Biol 243(2):245–251CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of Engineering and Computing Sciences Durham UniversityDurhamUK