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Evolutionary dynamics on any population structure


Evolution occurs in populations of reproducing individuals. The structure of a population can affect which traits evolve1,2. 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 neighbours3,4,5,6,7,8. 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 algorithm9. 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 times10,11 of random walks12. 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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Figure 1: Evolutionary games on weighted heterogeneous graphs.
Figure 2: Graphs that promote or hinder cooperation.
Figure 3: Rescuing cooperation by graph surgery.
Figure 4: Conditions for cooperation on 1.3 million random graphs.


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This work was supported by Office of Naval Research grant N00014-16-1-2914, the John Templeton Foundation, a gift from B. Wu and E. Larson, AFOSR grant FA9550-13-1-0097 (G.L.), the James S. McDonnell Foundation (B.F.), and the Center for Mathematical Sciences and Applications at Harvard University (B.A., Y.-T.C). We are grateful to S. R. Sundaresan and D. I. Rubenstein for providing data on zebra and wild ass networks, and for discussions.

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Authors and Affiliations



B.A., G.L., Y.-T.C. and M.A.N. conceived the project. B.A., G.L., Y.-T.C. and S.-T.Y. performed mathematical analysis. N.M., B.F. and G.L. performed numerical experiments and Monte Carlo simulations. B.A. and M.A.N. wrote the paper. All authors contributed to all aspects of the project.

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Correspondence to Benjamin Allen.

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Extended data figures and tables

Extended Data Figure 1 Further numerical and simulation results.

a, To assess accuracy of our results for empirically plausible selection strength, we performed Monte Carlo simulations with δ = 0.025 and c = 1. This corresponds to a fitness cost of 2.5%, which was determined to be the cost of cooperative behaviour in yeast43. Markers indicate population size times frequency of fixation for a particular value of b on a particular graph. Dashed lines indicate (b/c)* as calculated from equation (2). All graphs have size N = 100. Graphs are: Barabási–Albert (BA) with linking number m = 3, small world35 (SW) with initial connection distance d = 3 and edge creation probability p = 0.025, Klemm–Eguiluz39 (KE) with linking number m = 5 and deactivation parameter μ = 0.2, and Holme–Kim38 (HK) with linking number m = 2 and triad formation parameter P = 0.2. b, We computed (b/c)* for 4 × 104 large random graphs (sizes 300–1,000) using four random graph models: Erdös–Rényi34 (ER) with edge probability 0 < p < 0.25, Klemm–Eguiluz with linking number 3 ≤ m ≤ 5 and deactivation parameter 0 < μ < 0.15, Holme–Kim with linking number 2 ≤ m ≤ 4 and triad formation parameter 0 < P < 0.15, and a meta-network42 of shifted-linear preferential attachment networks40 (Island Barabási–Albert) with 0 < pinter < 0.25; see Methods for details. c, d, We computed the structure coefficient27 σ = ((b/c)* + 1)/((b/c)* − 1) for the same ensemble of random graphs as in Fig. 4 of the main text. Strategy A is favoured over strategy B under weak selection if σa + b > c + σd; see equation (3) of Methods. c, Plot of σ versus , which is the σ-value for a regular graph of the same mean degree . d, Plot of σ versus , which is the σ-value one would expect if the condition (as described in ref. 26) were exact. Here, is the expected degree of a neighbour of a randomly chosen vertex.

Extended Data Figure 2 The critical benefit–cost threshold for all graphs of size four.

There are six connected, unweighted graphs of size four. Of these, only the path graph has positive (b/c)*. Two others have infinite (b/c)* and three have negative (b/c)*. For size three (not shown), there are only two connected, unweighted graphs: the path, which has (b/c)* = ∞, and the triangle, which has (b/c)* = −2.

Extended Data Figure 3 The critical benefit–cost threshold for all graphs of size five.

There are 21 connected, unweighted graphs of size five. Critical ratios in the range 0 < (b/c)* < 30 are shown. Overall, seven of the (b/c)* values are positive, twelve are negative, and two are infinite.

Extended Data Figure 4 The critical benefit–cost threshold for all graphs of size six.

There are 112 connected, unweighted graphs of size six. Of these, 43 have positive (b/c)*, 65 have negative (b/c)*, and four have (b/c)* = ∞. Notably, there are graphs with the same degree sequence (for example, 3, 2, 2, 1, 1, 1) but different (b/c)*.

Extended Data Figure 5 Results for empirical networks.

The benefit–cost threshold (b/c)*, or equivalently the structure coefficient2,27 σ, gives the propensity of a population structure to support cooperative and/or Pareto-efficient behaviours. These values should be interpreted in terms of specific behaviours occurring in a population, and they depend on the network ontology (that is, the meaning of links). They can be used to facilitate comparisons across populations of similar species, or to predict consequences of changes in population structure. a, Unweighted social network of frequent associations in bottlenose dolphins (Tursiops spp.)44. b, Grooming interaction network in rhesus macaques (Macaca mulatta), weighted by grooming frequency45. c, Weighted network of group association in Grevy’s zebras (Equus grevyi)46. Preferred associations, which are statistically more frequent than random, are given weight 1. Other associations occurring at least once are given weight . d, Weighted network of group association in Asiatic wild asses (onagers)46, with the same weighting scheme as for the zebra network. For both zebra and wild ass, the unweighted networks, including every association ever observed, are dense and behave like well-mixed populations. By contrast, the weighted networks, which emphasize close ties, can promote cooperation. e, Table showing data from ad as well as a social network of family, self-reported friends, and co-workers as of 1971 from the Framingham Heart Study47,48, a Facebook ego-network49, and the co-authorship network for the General Relativity (gr) and Quantum Cosmology (qc) category of the arXiv preprint server50. Average degree is reported for unweighted graphs only; for weighted graphs it is unclear which notion of degree is most relevant. Note that large (b/c)* ratios, which correspond to σ values close to one, do not mean that cooperation is never favoured. Rather, the implication is that the network behaves similarly to a large well-mixed population, for which cooperation is favoured for any 2 × 2 game with a + b > c + d. The donation game does not satisfy this inequality, but other cooperative interactions do51,52.

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Allen, B., Lippner, G., Chen, YT. et al. Evolutionary dynamics on any population structure. Nature 544, 227–230 (2017).

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