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Nature. 2017 Apr 13;544(7649):227-230. doi: 10.1038/nature21723. Epub 2017 Mar 29.

Evolutionary dynamics on any population structure.

Allen B1,2,3, Lippner G3,4, Chen YT2,3,5, Fotouhi B2,6, Momeni N2,7, Yau ST3,8, Nowak MA2,8,9.

Author information

Department of Mathematics, Emmanuel College, Boston, Massachusetts, USA.
Program for Evolutionary Dynamics, Harvard University, Cambridge, Massachusetts, USA.
Center for Mathematical Sciences and Applications, Harvard University, Cambridge, Massachusetts, USA.
Department of Mathematics, Northeastern University, Boston, Massachusetts, USA.
Department of Mathematics, University of Tennessee, Knoxville, Tennessee, USA.
Institute for Quantitative Social Sciences, Harvard University, Cambridge, Massachusetts, USA.
Department of Electrical and Computer Engineering, McGill University, Montreal, Canada.
Department of Mathematics, Harvard University, Cambridge, Massachusetts, USA.
Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, Massachusetts, USA.


Evolution occurs in populations of reproducing individuals. The structure of a population can affect which traits evolve. Understanding evolutionary game dynamics in structured populations remains difficult. Mathematical results are known for special structures in which all individuals have the same number of neighbours. The general case, in which the number of neighbours can vary, has remained open. For arbitrary selection intensity, the problem is in a computational complexity class that suggests there is no efficient algorithm. Whether a simple solution for weak selection exists has remained unanswered. Here we provide a solution for weak selection that applies to any graph or network. Our method relies on calculating the coalescence times of random walks. We evaluate large numbers of diverse population structures for their propensity to favour cooperation. We study how small changes in population structure-graph surgery-affect evolutionary outcomes. We find that cooperation flourishes most in societies that are based on strong pairwise ties.

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