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.
Subscribe to Journal
Get full journal access for 1 year
only $3.90 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Rent or Buy article
Get time limited or full article access on ReadCube.
All prices are NET prices.
Liggett, T. M. Stochastic Interacting Systems: Contact, Voter and Exclusion Processes (Springer, Berlin, 1999)
Durrett, R. & Levin, S. A. The importance of being discrete (and spatial). Theor. Popul. Biol. 46, 363–394 (1994)
Moran, P. A. P. Random processes in genetics. Proc. Camb. Phil. Soc. 54, 60–71 (1958)
Durrett, R. A. Lecture Notes on Particle Systems & Percolation (Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, 1988)
Erdös, P. & Renyi, A. On the evolution of random graphs. Publ. Math. Inst. Hungarian Acad. Sci. 5, 17–61 (1960)
Barabasi, A. & Albert, R. Emergence of scaling in random networks. Science 286, 509–512 (1999)
Nagylaki, T. & Lucier, B. Numerical analysis of random drift in a cline. Genetics 94, 497–517 (1980)
Wright, S. Evolution in Mendelian populations. Genetics 16, 97–159 (1931)
Wright, S. The roles of mutation, inbreeding, crossbreeding and selection in evolution. Proc. 6th Int. Congr. Genet. 1, 356–366 (1932)
Fisher, R. A. & Ford, E. B. The “Sewall Wright Effect”. Heredity 4, 117–119 (1950)
Barton, N. The probability of fixation of a favoured allele in a subdivided population. Genet. Res. 62, 149–158 (1993)
Whitlock, M. Fixation probability and time in subdivided populations. Genetics 164, 767–779 (2003)
Nowak, M. A. & May, R. M. The spatial dilemmas of evolution. Int. J. Bifurcation Chaos 3, 35–78 (1993)
Hauert, C. & Doebeli, M. Spatial structure often inhibits the evolution of cooperation in the snowdrift game. Nature 428, 643–646 (2004)
Hofbauer, J. & Sigmund, K. Evolutionary Games and Population Dynamics (Cambridge Univ. Press, Cambridge, 1998)
Maruyama, T. Effective number of alleles in a subdivided population. Theor. Popul. Biol. 1, 273–306 (1970)
Slatkin, M. Fixation probabilities and fixation times in a subdivided population. Evolution 35, 477–488 (1981)
Ebel, H. & Bornholdt, S. Coevolutionary games on networks. Phys. Rev. E. 66, 056118 (2002)
Abramson, G. & Kuperman, M. Social games in a social network. Phys. Rev. E. 63, 030901(R) (2001)
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)
Neuhauser, C. Mathematical challenges in spatial ecology. Not. AMS 48, 1304–1314 (2001)
Pulliam, H. R. Sources, sinks, and population regulation. Am. Nat. 132, 652–661 (1988)
Hassell, M. P., Comins, H. N. & May, R. M. Species coexistence and self-organizing spatial dynamics. Nature 370, 290–292 (1994)
Reya, T., Morrison, S. J., Clarke, M. & Weissman, I. L. Stem cells, cancer, and cancer stem cells. Nature 414, 105–111 (2001)
Skyrms, B. & Pemantle, R. A dynamic model of social network formation. Proc. Nat. Acad. Sci. USA 97, 9340–9346 (2000)
Jackson, M. O. & Watts, A. On the formation of interaction networks in social coordination games. Games Econ. Behav. 41, 265–291 (2002)
Asavathiratham, C., Roy, S., Lesieutre, B. & Verghese, G. The influence model. IEEE Control Syst. Mag. 21, 52–64 (2001)
Newman, M. E. J. The structure of scientific collaboration networks. Proc. Natl Acad. Sci. USA 98, 404–409 (2001)
Boyd, S., Diaconis, P. & Xiao, L. Fastest mixing Markov chain on a graph. SIAM Rev. 46, 667–689 (2004)
Nakamaru, M., Matsuda, H. & Iwasa, Y. The evolution of cooperation in a lattice-structured population. J. Theor. Biol. 184, 65–81 (1997)
Bala, V. & Goyal, S. A noncooperative model of network formation. Econometrica 68, 1181–1229 (2000)
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.
The authors declare that they have no competing financial interests.
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)
About this article
Cite this article
Lieberman, E., Hauert, C. & Nowak, M. Evolutionary dynamics on graphs. Nature 433, 312–316 (2005). https://doi.org/10.1038/nature03204
Dynamic Games and Applications (2020)
European Journal of Control (2020)
Combining local and global evolutionary trajectories of brain–behaviour relationships through game theory
European Journal of Neuroscience (2020)
Scientific Reports (2020)
IEEE Access (2020)