Evolutionary dynamics on graphs (original) (raw)

Nature volume 433, pages 312–316 (2005)Cite this article

Abstract

Evolutionary dynamics have been traditionally studied in the context of homogeneous or spatially extended populations1,2,3,4. Here we generalize population structure by arranging individuals on a graph. Each vertex represents an individual. The weighted edges denote reproductive rates which govern how often individuals place offspring into adjacent vertices. The homogeneous population, described by the Moran process3, is the special case of a fully connected graph with evenly weighted edges. Spatial structures are described by graphs where vertices are connected with their nearest neighbours. We also explore evolution on random and scale-free networks5,6,7. We determine the fixation probability of mutants, and characterize those graphs for which fixation behaviour is identical to that of a homogeneous population7. Furthermore, some graphs act as suppressors and others as amplifiers of selection. It is even possible to find graphs that guarantee the fixation of any advantageous mutant. We also study frequency-dependent selection and show that the outcome of evolutionary games can depend entirely on the structure of the underlying graph. Evolutionary graph theory has many fascinating applications ranging from ecology to multi-cellular organization and economics.

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References

  1. Liggett, T. M. Stochastic Interacting Systems: Contact, Voter and Exclusion Processes (Springer, Berlin, 1999)
    Book Google Scholar
  2. Durrett, R. & Levin, S. A. The importance of being discrete (and spatial). Theor. Popul. Biol. 46, 363–394 (1994)
    Article Google Scholar
  3. Moran, P. A. P. Random processes in genetics. Proc. Camb. Phil. Soc. 54, 60–71 (1958)
    Article ADS MathSciNet Google Scholar
  4. Durrett, R. A. Lecture Notes on Particle Systems & Percolation (Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, 1988)
    MATH Google Scholar
  5. Erdös, P. & Renyi, A. On the evolution of random graphs. Publ. Math. Inst. Hungarian Acad. Sci. 5, 17–61 (1960)
    MathSciNet MATH Google Scholar
  6. Barabasi, A. & Albert, R. Emergence of scaling in random networks. Science 286, 509–512 (1999)
    Article ADS MathSciNet CAS Google Scholar
  7. Nagylaki, T. & Lucier, B. Numerical analysis of random drift in a cline. Genetics 94, 497–517 (1980)
    MathSciNet CAS PubMed PubMed Central Google Scholar
  8. Wright, S. Evolution in Mendelian populations. Genetics 16, 97–159 (1931)
    CAS PubMed PubMed Central Google Scholar
  9. Wright, S. The roles of mutation, inbreeding, crossbreeding and selection in evolution. Proc. 6th Int. Congr. Genet. 1, 356–366 (1932)
    Google Scholar
  10. Fisher, R. A. & Ford, E. B. The “Sewall Wright Effect”. Heredity 4, 117–119 (1950)
    Article CAS Google Scholar
  11. Barton, N. The probability of fixation of a favoured allele in a subdivided population. Genet. Res. 62, 149–158 (1993)
    Article Google Scholar
  12. Whitlock, M. Fixation probability and time in subdivided populations. Genetics 164, 767–779 (2003)
    PubMed PubMed Central Google Scholar
  13. Nowak, M. A. & May, R. M. The spatial dilemmas of evolution. Int. J. Bifurcation Chaos 3, 35–78 (1993)
    Article ADS MathSciNet Google Scholar
  14. Hauert, C. & Doebeli, M. Spatial structure often inhibits the evolution of cooperation in the snowdrift game. Nature 428, 643–646 (2004)
    Article ADS CAS Google Scholar
  15. Hofbauer, J. & Sigmund, K. Evolutionary Games and Population Dynamics (Cambridge Univ. Press, Cambridge, 1998)
    Book Google Scholar
  16. Maruyama, T. Effective number of alleles in a subdivided population. Theor. Popul. Biol. 1, 273–306 (1970)
    Article MathSciNet CAS Google Scholar
  17. Slatkin, M. Fixation probabilities and fixation times in a subdivided population. Evolution 35, 477–488 (1981)
    Article Google Scholar
  18. Ebel, H. & Bornholdt, S. Coevolutionary games on networks. Phys. Rev. E. 66, 056118 (2002)
    Article ADS Google Scholar
  19. Abramson, G. & Kuperman, M. Social games in a social network. Phys. Rev. E. 63, 030901(R) (2001)
    Article ADS Google Scholar
  20. Tilman, D. & Karieva, P. (eds) Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions (Monographs in Population Biology, Princeton Univ. Press, Princeton, 1997)
  21. Neuhauser, C. Mathematical challenges in spatial ecology. Not. AMS 48, 1304–1314 (2001)
    MathSciNet MATH Google Scholar
  22. Pulliam, H. R. Sources, sinks, and population regulation. Am. Nat. 132, 652–661 (1988)
    Article Google Scholar
  23. Hassell, M. P., Comins, H. N. & May, R. M. Species coexistence and self-organizing spatial dynamics. Nature 370, 290–292 (1994)
    Article ADS Google Scholar
  24. Reya, T., Morrison, S. J., Clarke, M. & Weissman, I. L. Stem cells, cancer, and cancer stem cells. Nature 414, 105–111 (2001)
    Article ADS CAS Google Scholar
  25. Skyrms, B. & Pemantle, R. A dynamic model of social network formation. Proc. Nat. Acad. Sci. USA 97, 9340–9346 (2000)
    Article ADS CAS Google Scholar
  26. Jackson, M. O. & Watts, A. On the formation of interaction networks in social coordination games. Games Econ. Behav. 41, 265–291 (2002)
    Article MathSciNet Google Scholar
  27. Asavathiratham, C., Roy, S., Lesieutre, B. & Verghese, G. The influence model. IEEE Control Syst. Mag. 21, 52–64 (2001)
    Article Google Scholar
  28. Newman, M. E. J. The structure of scientific collaboration networks. Proc. Natl Acad. Sci. USA 98, 404–409 (2001)
    Article ADS MathSciNet CAS Google Scholar
  29. Boyd, S., Diaconis, P. & Xiao, L. Fastest mixing Markov chain on a graph. SIAM Rev. 46, 667–689 (2004)
    Article ADS MathSciNet Google Scholar
  30. Nakamaru, M., Matsuda, H. & Iwasa, Y. The evolution of cooperation in a lattice-structured population. J. Theor. Biol. 184, 65–81 (1997)
    Article CAS Google Scholar
  31. Bala, V. & Goyal, S. A noncooperative model of network formation. Econometrica 68, 1181–1229 (2000)
    Article MathSciNet Google Scholar

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Acknowledgements

The Program for Evolutionary Dynamics is sponsored by J. Epstein. E.L. is supported by a National Defense Science and Engineering Graduate Fellowship. C.H. is grateful to the Swiss National Science Foundation. We are indebted to M. Brenner for many discussions.

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Authors and Affiliations

  1. Program for Evolutionary Dynamics, Departments of Organismic and Evolutionary Biology, Mathematics, and Applied Mathematics, Harvard University, Cambridge, Massachusetts, 02138, USA
    Erez Lieberman, Christoph Hauert & Martin A. Nowak
  2. Harvard-MIT Division of Health Sciences and Technology, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA
    Erez Lieberman
  3. Department of Zoology, University of British Columbia, Vancouver, British Columbia, V6T 1Z4, Canada
    Christoph Hauert

Authors

  1. Erez Lieberman
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  2. Christoph Hauert
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  3. Martin A. Nowak
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Corresponding author

Correspondence toErez Lieberman.

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The authors declare that they have no competing financial interests.

Supplementary information

Supplementary Notes

These Supplementary Notes outline the derivations of the major results stated in the main text and provide a discussion of their robustness. It contains a sketch of the derivations of equation (1) for circulations and equation (2) for superstars, and also gives a brief discussion of complexity results for frequency-dependent selection and the computation underlying results for directed cycles. This closes with a discussion of assumptions about mutation rate and the interpretations of fitness that these results can accommodate. (PDF 118 kb)

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Lieberman, E., Hauert, C. & Nowak, M. Evolutionary dynamics on graphs.Nature 433, 312–316 (2005). https://doi.org/10.1038/nature03204

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