Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Strategy evolution on higher-order networks

Nature Computational Science volume 4, pages 274–284 (2024)Cite this article

Subjects

Abstract

Cooperation is key to prosperity in human societies. Population structure is well understood as a catalyst for cooperation, where research has focused on pairwise interactions. But cooperative behaviors are not simply dyadic, and they often involve coordinated behavior in larger groups. Here we develop a framework to study the evolution of behavioral strategies in higher-order population structures, which include pairwise and multi-way interactions. We provide an analytical treatment of when cooperation will be favored by higher-order interactions, accounting for arbitrary spatial heterogeneity and nonlinear rewards for cooperation in larger groups. Our results indicate that higher-order interactions can act to promote the evolution of cooperation across a broad range of networks, in public goods games. Higher-order interactions consistently provide an advantage for cooperation when interaction hyper-networks feature multiple conjoined communities. Our analysis provides a systematic account of how higher-order interactions modulate the evolution of prosocial traits.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Evolutionary dynamics with higher-order interactions.
Fig. 2: The evolution of cooperative behavior depends on the number of players in PGGs and the strength of nonlinearity in their contributions.
Fig. 3: Higher-order interactions can facilitate cooperation.
Fig. 4: Intuition for how higher-order interactions promote cooperation.
Fig. 5: Higher-order interactions facilitate cooperation in multi-clique networks.
Fig. 6: Effects of higher-order interactions for empirical social networks.

Similar content being viewed by others

Data availability

All empirical network datasets used in this paper are freely and publicly available at https://icon.colorado.edu/#!/networks (see refs. 55,56,57,58). Source data are provided with this paper.

Code availability

All numerical calculations were performed in MATLAB R2023a. All data analysis were performed in Python 3.10. All computer code developed in this study has been deposited into a publicly available GitHub repository at https://github.com/anzhisheng/Higher-order-interactions (ref. 75).

References

  1. Bavel, JayJ. Van et al. Using social and behavioural science to support COVID-19 pandemic response. Nat. Hum. Behav. 4, 460–471 (2020).

    Google Scholar 

  2. Jacquet, J. et al. Intra-and intergenerational discounting in the climate game. Nat. Clim. Change 3, 1025–1028 (2013).

    Google Scholar 

  3. Tilman, A. R., Plotkin, J. B. & Akçay, E. Evolutionary games with environmental feedbacks. Nat. Commun. 11, 915 (2020).

    Google Scholar 

  4. Perc, Matjaž. et al. Statistical physics of human cooperation. Phys. Rep. 687, 1–51 (2017).

    MathSciNet  Google Scholar 

  5. Nowak, M. A. & May, R. M. Evolutionary games and spatial chaos. Nature 359, 826–829 (1992).

    Google Scholar 

  6. Santos, F. C. & Pacheco, J. M. Scale-free networks provide a unifying framework for the emergence of cooperation. Phys. Rev. Lett. 95, 098104 (2005).

    Google Scholar 

  7. Nowak, M. A. Five rules for the evolution of cooperation. Science 314, 1560–1563 (2006).

    Google Scholar 

  8. Ohtsuki, H., Hauert, C., Lieberman, E. & Nowak, M. A. A simple rule for the evolution of cooperation on graphs and social networks. Nature 441, 502–505 (2006).

    Google Scholar 

  9. Lehmann, L., Keller, L. & Sumpter, DavidJ. T. The evolution of helping and harming on graphs: the return of the inclusive fitness effect. J. Evol. Biol. 20, 2284–2295 (2007).

    Google Scholar 

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

    Google Scholar 

  11. Nowak, M. A., Sasaki, A., Taylor, C. & Fudenberg, D. Emergence of cooperation and evolutionary stability in finite populations. Nature 428, 646–650 (2004).

    Google Scholar 

  12. Taylor, P. D., Day, T. & Wild, G. Evolution of cooperation in a finite homogeneous graph. Nature 447, 469–472 (2007).

    Google Scholar 

  13. Tarnita, C. E., Ohtsuki, H., Antal, T., Fu, F. & Nowak, M. A. Strategy selection in structured populations. J. Theor. Biol. 259, 570–581 (2009).

    MathSciNet  Google Scholar 

  14. Su, Q., Allen, B. & Plotkin, J. B. Evolution of cooperation with asymmetric social interactions. Proc. Natl Acad. Sci. USA 119, e2113468118 (2022).

    Google Scholar 

  15. Sheng, A., Li, A. & Wang, L. Evolutionary dynamics on sequential temporal networks. PLoS Comput. Biol. 19, e1011333 (2023).

    Google Scholar 

  16. Xia, C., Wang, J., Perc, Matjaž. & Wang, Z. Reputation and reciprocity. Phys. Life Rev. 46, 8–45 (2023).

    Google Scholar 

  17. Rousset, François & Billiard, S. A theoretical basis for measures of kin selection in subdivided populations: finite populations and localized dispersal. J. Evol. Biol. 13, 814–825 (2000).

    Google Scholar 

  18. Rousset, F. Genetic Structure and Selection in Subdivided Populations Vol. 40 (Princeton Univ. Press, 2004).

  19. Santos, F. C., Santos, M. D. & Pacheco, J. M. Social diversity promotes the emergence of cooperation in public goods games. Nature 454, 213–216 (2008).

    Google Scholar 

  20. Perc, Matjaž., Gómez-Gardenes, Jesús, Szolnoki, A., Floría, L. M. & Moreno, Y. Evolutionary dynamics of group interactions on structured populations: a review. J. R. Soc. Interface 10, 20120997 (2013).

    Google Scholar 

  21. Gokhale, C. S. & Traulsen, A. Evolutionary multiplayer games. Dyn. Games Appl. 4, 468–488 (2014).

    MathSciNet  Google Scholar 

  22. Mullon, C. & Lehmann, L. The robustness of the weak selection approximation for the evolution of altruism against strong selection. J. Evol. Biol. 27, 2272–2282 (2014).

    Google Scholar 

  23. Peña, J., Nöldeke, G. & Lehmann, L. Evolutionary dynamics of collective action in spatially structured populations. J. Theor. Biol. 382, 122–136 (2015).

    MathSciNet  Google Scholar 

  24. Van Cleve, J. Social evolution and genetic interactions in the short and long term. Theor. Popul. Biol. 103, 2–26 (2015).

    Google Scholar 

  25. Benson, A. R., Gleich, D. F. & Leskovec, J. Higher-order organization of complex networks. Science 353, 163–166 (2016).

    Google Scholar 

  26. Battiston, F. et al. Networks beyond pairwise interactions: structure and dynamics. Phys. Rep. 874, 1–92 (2020).

    MathSciNet  Google Scholar 

  27. Battiston, F. et al. The physics of higher-order interactions in complex systems. Nat. Phys. 17, 1093–1098 (2021).

    Google Scholar 

  28. Hauert, C., Michor, F., Nowak, M. A. & Doebeli, M. Synergy and discounting of cooperation in social dilemmas. J. Theor. Biol. 239, 195–202 (2006).

    MathSciNet  Google Scholar 

  29. Peña, J. & Nöldeke, G. Cooperative dilemmas with binary actions and multiple players. Dyn. Games Appl. 13, 1156–1193 (2023).

    MathSciNet  Google Scholar 

  30. Levine, J. M., Bascompte, J., Adler, P. B. & Allesina, S. Beyond pairwise mechanisms of species coexistence in complex communities. Nature 546, 56–64 (2017).

    Google Scholar 

  31. Benson, A. R., Abebe, R., Schaub, M. T., Jadbabaie, A. & Kleinberg, J. Simplicial closure and higher-order link prediction. Proc. Natl Acad. Sci. USA 115, E11221–E11230 (2018).

    Google Scholar 

  32. Ganmor, E., Segev, R. & Schneidman, E. Sparse low-order interaction network underlies a highly correlated and learnable neural population code. Proc. Natl Acad. Sci. USA 108, 9679–9684 (2011).

    Google Scholar 

  33. Milojević, Staša Principles of scientific research team formation and evolution. Proc. Natl Acad. Sci. USA 111, 3984–3989 (2014).

    Google Scholar 

  34. Gambuzza, L. V. et al. Stability of synchronization in simplicial complexes. Nat. Commun. 12, 1255 (2021).

    Google Scholar 

  35. Grilli, J., Barabás, György, Michalska-Smith, M. J. & Allesina, S. Higher-order interactions stabilize dynamics in competitive network models. Nature 548, 210–213 (2017).

    Google Scholar 

  36. Iacopini, I., Petri, G., Barrat, A. & Latora, V. Simplicial models of social contagion. Nat. Commun. 10, 2485 (2019).

    Google Scholar 

  37. Peña, J. & Nöldeke, G. Group size effects in social evolution. J. Theor. Biol. 457, 211–220 (2018).

    MathSciNet  Google Scholar 

  38. Peña, J. Group-size diversity in public goods games. Evolution 66, 623–636 (2012).

    Google Scholar 

  39. Peña, J., Wu, B., Arranz, J. & Traulsen, A. Evolutionary games of multiplayer cooperation on graphs. PLoS Comput. Biol. 12, e1005059 (2016).

    Google Scholar 

  40. Li, A., Broom, M., Du, J. & Wang, L. Evolutionary dynamics of general group interactions in structured populations. Phys. Rev. E 93, 022407 (2016).

    MathSciNet  Google Scholar 

  41. Su, Q., Zhou, L. & Wang, L. Evolutionary multiplayer games on graphs with edge diversity. PLoS Comput. Biol. 15, e1006947 (2019).

    Google Scholar 

  42. Broom, M. & Rychtář, J. A general framework for analysing multiplayer games in networks using territorial interactions as a case study. J. Theor. Biol. 302, 70–80 (2012).

    MathSciNet  Google Scholar 

  43. Gomez-Gardenes, J., Romance, M., Criado, R., Vilone, D. & Sánchez, A. Evolutionary games defined at the network mesoscale: the public goods game. Chaos 21, 016113 (2011).

    MathSciNet  Google Scholar 

  44. Peña, J. & Rochat, Y. Bipartite graphs as models of population structures in evolutionary multiplayer games. PLoS ONE 7, e44514–e44514 (2012).

    Google Scholar 

  45. Alvarez-Rodriguez, U. et al. Evolutionary dynamics of higher-order interactions in social networks. Nat. Hum. Behav. 5, 586–595 (2021).

    Google Scholar 

  46. Su, Q., Li, A., Wang, L. & Stanley, H. E. Spatial reciprocity in the evolution of cooperation. Proc. R. Soc. B 286, 20190041 (2019).

    Google Scholar 

  47. Wu, B., Altrock, P. M., Wang, L. & Traulsen, A. Universality of weak selection. Phys. Rev. E 82, 046106 (2010).

    Google Scholar 

  48. McAvoy, A. & Allen, B. Fixation probabilities in evolutionary dynamics under weak selection. J. Math. Biol. 82, 14 (2021).

    MathSciNet  Google Scholar 

  49. Gokhale, C. S. & Traulsen, A. Evolutionary games in the multiverse. Proc. Natl Acad. Sci. USA 107, 5500–5504 (2010).

    MathSciNet  Google Scholar 

  50. Liu, R. & Masuda, N. Fixation dynamics on hypergraphs. PLoS Comput. Biol. 19, e1011494 (2023).

    Google Scholar 

  51. Peña, J. & Nöldeke, G. Variability in group size and the evolution of collective action. J. Theor. Biol. 389, 72–82 (2016).

    MathSciNet  Google Scholar 

  52. Fotouhi, B., Momeni, N., Allen, B. & Nowak, M. A. Conjoining uncooperative societies facilitates evolution of cooperation. Nat. Hum. Behav. 2, 492–499 (2018).

    Google Scholar 

  53. Watts, D. J. & Strogatz, S. H. Collective dynamics of ‘small-world’ networks. Nature 393, 440–442 (1998).

    Google Scholar 

  54. Barabási, Albert-L. ászló & Albert, R. éka Emergence of scaling in random networks. Science 286, 509–512 (1999).

    MathSciNet  Google Scholar 

  55. Rogers, E. M. & Kincaid, D. L. Communication Networks: toward A New Paradigm for Research (Free Press, 1981).

  56. Zachary, W. W. An information flow model for conflict and fission in small groups. J. Anthropol. Res. 33, 452–473 (1977).

    Google Scholar 

  57. Davis, A., Gardner, B. B. & Gardner, M. R. Deep South: A Social Anthropological Study of Caste and Class (Univ. South Carolina Press, 2009).

  58. Anderson, C. J., Wasserman, S. & Crouch, B. A p* primer: logit models for social networks. Soc. Netw. 21, 37–66 (1999).

    Google Scholar 

  59. Clauset, A., Newman, MarkE. J. & Moore, C. Finding community structure in very large networks. Phys. Rev. E 70, 066111 (2004).

    Google Scholar 

  60. Wang, J., Fu, F., Wu, T. & Wang, L. Emergence of social cooperation in threshold public goods games with collective risk. Phys. Rev. E 80, 016101 (2009).

    Google Scholar 

  61. Wu, T., Fu, F. & Wang, L. Partner selections in public goods games with constant group size. Phys. Rev. E 80, 026121 (2009).

    Google Scholar 

  62. Szabó, György & Hauert, C. Phase transitions and volunteering in spatial public goods games. Phys. Rev. Lett. 89, 118101 (2002).

    Google Scholar 

  63. Szolnoki, A. & Perc, Matjaž. Group-size effects on the evolution of cooperation in the spatial public goods game. Phys. Rev. E 84, 047102 (2011).

    Google Scholar 

  64. Souza, M. O., Pacheco, J. M. & Santos, F. C. Evolution of cooperation under n-person snowdrift games. J. Theor. Biol. 260, 581–588 (2009).

    MathSciNet  Google Scholar 

  65. Civilini, A., Anbarci, N. & Latora, V. Evolutionary game model of group choice dilemmas on hypergraphs. Phys. Rev. Lett. 127, 268301 (2021).

    MathSciNet  Google Scholar 

  66. Allen, B., Lippner, G. & Nowak, M. A. Evolutionary games on isothermal graphs. Nat. Commun. 10, 5107 (2019).

    Google Scholar 

  67. McAvoy, A. & Wakeley, J. Evaluating the structure-coefficient theorem of evolutionary game theory. Proc. Natl Acad. Sci. USA 119, e2119656119 (2022).

    MathSciNet  Google Scholar 

  68. Wu, B., García, Julián, Hauert, C. & Traulsen, A. Extrapolating weak selection in evolutionary games. PLoS Comput. Biol. 9, e1003381 (2013).

    Google Scholar 

  69. Fortunato, S. Community detection in graphs. Phys. Rep. 486, 75–174 (2010).

    MathSciNet  Google Scholar 

  70. Fortunato, S. & Hric, D. Community detection in networks: a user guide. Phys. Rep. 659, 1–44 (2016).

    MathSciNet  Google Scholar 

  71. Sheng, A., Su, Q., Li, A., Wang, L. & Plotkin, J. B. Constructing temporal networks with bursty activity patterns. Nat. Commun. 14, 7311 (2023).

    Google Scholar 

  72. Kumar, A., Chowdhary, S., Capraro, V. & Perc, Matjaž. Evolution of honesty in higher-order social networks. Phys. Rev. E 104, 054308 (2021).

    MathSciNet  Google Scholar 

  73. McAvoy, A. & Hauert, C. Asymmetric evolutionary games. PLoS Comput. Biol. 11, e1004349 (2015).

    Google Scholar 

  74. McAvoy, A., Allen, B. & Nowak, M. A. Social goods dilemmas in heterogeneous societies. Nat. Hum. Behav. 4, 819–831 (2020).

    Google Scholar 

  75. Initial release of code for research paper ‘Strategy evolution on higher-order networks’. Code Ocean https://doi.org/10.24433/CO.7059083.v2 (2024).

Download references

Acknowledgements

We thank A. McAvoy for useful comments and discussions. A.S. acknowledges support from the China Scholarship Council (no. 202206010147). Q.S. acknowledges support from Shanghai Pujiang Program (no. 23PJ1405500). L.W. acknowledges support from the National Natural Science Foundation of China (no. 62036002). J.B.P. acknowledges support from the Simons Foundation Math+X grant to the University of Pennsylvania, and from the John Templeton Foundation (grant #62281).

Author information

Authors and Affiliations

  1. Center for Systems and Control, College of Engineering, Peking University, Beijing, China

    Anzhi Sheng & Long Wang

  2. Department of Biology, University of Pennsylvania, Philadelphia, PA, USA

    Anzhi Sheng & Joshua B. Plotkin

  3. Department of Automation, Shanghai Jiao Tong University, Shanghai, China

    Qi Su

  4. Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai, China

    Qi Su

  5. Shanghai Engineering Research Center of Intelligent Control and Management, Shanghai, China

    Qi Su

  6. Center for Multi-Agent Research, Institute for Artificial Intelligence, Peking University, Beijing, China

    Long Wang

  7. Center for Mathematical Biology, University of Pennsylvania, Philadelphia, PA, USA

    Joshua B. Plotkin

Authors
  1. Anzhi Sheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qi Su
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Long Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Joshua B. Plotkin
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

A.S., Q.S., L.W. and J.B.P. conceived the project. A.S. derived analytical results and performed numerical calculations. A.S., Q.S., L.W. and J.B.P. analyzed the data. A.S., Q.S. and J.B.P. wrote the main text with input from L.W. A.S. wrote the Supplementary Information.

Corresponding authors

Correspondence to Qi Su, Long Wang or Joshua B. Plotkin.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Computational Science thanks Bin Wu and the other anonymous reviewer(s) for their contribution to the peer review of this work. Primary Handling Editor: Ananya Rastogi, in collaboration with the Nature Computational Science team.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data

Extended Data Fig. 1 Evolution of cooperation on a variety of higher-order networks.

Cooperation is typically facilitated by higher-order interactions in diverse networks with multiple cliques. a, There are 93 networks that have a size of N = 6 and contain at least one triangle. Two examples are illustrated in the upper panel. For every single network, the critical benefit-to-cost ratios for purely pairwise interactions \({(b/c)}_{(2)}^{* }\) and for higher-order interactions \({(b/c)}_{(2,3)}^{* }\) are calculated, and their difference measures how higher-order interactions influence the evolution of cooperation: \({(b/c)}_{(2)}^{* }-{(b/c)}_{(2,3)}^{* } > 0\) indicates that higher-order interactions facilitate cooperation. For δ2 = 1 and δ2 = 0, the red bars show that higher-order interactions facilitate cooperation for only 6 out of the 93 6-node networks. But when we conjoin two such single networks via a random link (averaging over link placement), then higher-order interactions consistently facilitate cooperation (blue bars). b, Three representative classes of random networks: random regular networks (RR), Watts-Strogatz small-world networks with rewiring probability p = 0.3 (SW)53, and Barabási-Albert scale-free networks (SF)54. For each class, we generated 1,000 networks with size N sampled uniformly from [8, 15] and with average degree sampled in [4,N]. The red bars show the effects of higher-order interactions in these single networks (which facilitate cooperation in less than 50% of cases), whereas the blue bars show the effects of higher-order interactions on two-clique networks generated by connecting two identical networks via the node of highest degree: higher-order interactions facilitate cooperation for all such conjoined networks.

Source data

Supplementary information

Supplementary Information

Supplementary Sections 1–5, Figs. 1–14 and Table 1.

Reporting Summary

Source data

Source Data Fig. 2

Statistical source data.

Source Data Fig. 3

Statistical source data.

Source Data Fig. 5

Statistical source data.

Source Data Extended Data Fig. 1

Numerical data.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sheng, A., Su, Q., Wang, L. et al. Strategy evolution on higher-order networks. Nat Comput Sci 4, 274–284 (2024). https://doi.org/10.1038/s43588-024-00621-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s43588-024-00621-8

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing