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Strategy evolution on higher-order networks

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

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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.

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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.

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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).

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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

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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.

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

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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.

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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.

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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

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