Evolutionary dynamics on graphs

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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Figure 1: Models of evolution.
Figure 2: Isothermal graphs, and, more generally, circulations, have fixation behaviour identical to the Moran process.
Figure 3: Selection amplifiers have remarkable symmetry properties.
Figure 4: Simulation results showing the likelihood of mutant fixation.
Figure 5: Evolutionary games on directed cycles for four different orientations.

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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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Correspondence to Erez Lieberman.

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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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