Abstract
We live and cooperate in networks. However, links in networks only allow for pairwise interactions, thus making the framework suitable for dyadic games, but not for games that are played in larger groups. Here, we study the evolutionary dynamics of a public goods game in social systems with higher-order interactions. First, we show that the game on uniform hypergraphs corresponds to the replicator dynamics in the well-mixed limit, providing a formal theoretical foundation to study cooperation in networked groups. Second, we unveil how the presence of hubs and the coexistence of interactions in groups of different sizes affects the evolution of cooperation. Finally, we apply the proposed framework to extract the actual dependence of the synergy factor on the size of a group from real-world collaboration data in science and technology. Our work provides a way to implement informed actions to boost cooperation in social groups.
This is a preview of subscription content, access via your institution
Access options
Access Nature and 54 other Nature Portfolio journals
Get Nature+, our best-value online-access subscription
$29.99 / 30 days
cancel any time
Subscribe to this journal
Receive 12 digital issues and online access to articles
$119.00 per year
only $9.92 per issue
Buy this article
- Purchase on Springer Link
- Instant access to full article PDF
Prices may be subject to local taxes which are calculated during checkout
Similar content being viewed by others
Data availability
The APS dataset is provided by the APS at: https://journals.aps.org/datasets.
Code availability
Custom code that supports the findings of this study is available from the corresponding author upon request.
References
Nowak, M. A. & Highfield, R. SuperCooperators: Altruism, Evolution, and Why We Need Each Other to Succeed (Free Press, 2011).
Santos, F. C., Rodrigues, J. F. & Pacheco, J. M. Graph topology plays a determinant role in the evolution of cooperation. Proc. R. Soc. B 273, 51–55 (2006).
Rand, D. G., Arbesman, S. & Christakis, N. A. Dynamic social networks promote cooperation in experiments with humans. Proc. Natl Acad. Sci. USA 108, 19193–19198 (2011).
Hrdy, S. B. et al. Mothers and Others: The Evolutionary Origins of Mutual Understanding (Harvard Univ. Press, 2011).
Henrich, J. et al. In search of homo economicus: behavioral experiments in 15 small-scale societies. Am. Econ. Rev. 91, 73–78 (2001).
Nowak, M. A. Five rules for the evolution of cooperation. Science 314, 1560–1563 (2006).
Henrich, N. & Henrich, J. P. Why Humans Cooperate: A Cultural and Evolutionary Explanation (Oxford Univ. Press, 2007).
Rand, D. G. & Nowak, M. A. Human cooperation. Trends Cognit. Sci. 17, 413–425 (2013).
Kraft-Todd, G., Yoeli, E., Bhanot, S. & Rand, D. Promoting cooperation in the field. Curr. Opin. Behav. Sci. 3, 96–101 (2015).
Perc, M. et al. Statistical physics of human cooperation. Phys. Rep. 687, 1–51 (2017).
Jackson, M. O. & Zenou, Y. Economic Analyses of Social Networks (Edward Elgar Publishing, 2013).
Weibull, J. W. Evolutionary Game Theory (MIT Press, 1995).
Hofbauer, J. & Sigmund, K. Evolutionary Games and Population Dynamics (Cambridge Univ. Press, 1998).
Nowak, M. A. Evolutionary Dynamics (Harvard Univ. Press, 2006).
Axelrod, R. The Evolution of Cooperation (Basic Books, 1984).
Nowak, M. A. & May, R. M. Evolutionary games and spatial chaos. Nature 359, 826–829 (1992).
Santos, F. C. & Pacheco, J. M. Scale-free networks provide a unifying framework for the emergence of cooperation. Phys. Rev. Lett. 95, 098104 (2005).
Santos, F. C., Pacheco, J. M. & Lenaerts, T. Evolutionary dynamics of social dilemmas in structured heterogeneous populations. Proc. Natl Acad. Sci. USA 103, 3490–3494 (2006).
Gómez-Gardeñes, J., Campillo, M., Floría, L. M. & Moreno, Y. Dynamical organization of cooperation in complex networks. Phys. Rev. Lett. 98, 108103 (2007).
Tarnita, C. E., Antal, T., Ohtsuki, H. & Nowak, M. A. Evolutionary dynamics in set structured populations. Proc. Natl Acad. Sci. USA 106, 8601–8604 (2009).
Fotouhi, B., Momeni, N., Allen, B. & Nowak, M. A. Evolution of cooperation on large networks with community structure. J. R. Soc. Interface 16, 20180677 (2019).
Wang, Z., Szolnoki, A. & Perc, M. Evolution of public cooperation on interdependent networks: The impact of biased utility functions. Europhys. Lett. 97, 48001 (2012).
Gómez-Gardeñes, J., Reinares, I., Arenas, A. & Floría, L. M. Evolution of cooperation in multiplex networks. Sci. Rep. 2, 620 (2012).
Gómez-Gardeñes, J., Gracia-Lázaro, C., Floría, L. M. & Moreno, Y. Evolutionary dynamics on interdependent populations. Phys. Rev. E 86, 056113 (2012).
Wang, Z., Szolnoki, A. & Perc, M. Interdependent network reciprocity in evolutionary games. Sci. Rep. 3, 1183 (2013).
Wang, Z., Wang, L. & Perc, M. Degree mixing in multilayer networks impedes the evolution of cooperation. Phys. Rev. E 89, 052813 (2014).
Battiston, F., Perc, M. & Latora, V. Determinants of public cooperation in multiplex networks. New J. Phys. 19, 073017 (2017).
Fu, F. & Chen, X. Leveraging statistical physics to improve understanding of cooperation in multiplex networks. New J. Phys. 19, 071002 (2017).
Fotouhi, B., Momeni, N., Allen, B. & Nowak, M. A. Conjoining uncooperative societies facilitates evolution of cooperation. Nat. Human Behav. 2, 492 (2018).
Lieberman, E., Hauert, C. & Nowak, M. A. Evolutionary dynamics on graphs. Nature 433, 312–316 (2005).
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).
Allen, B. et al. Evolutionary dynamics on any population structure. Nature 544, 227–230 (2017).
Archetti, M. & Scheuring, I. Game theory of public goods in one-shot social dilemmas without assortment. J. Theor. Biol. 299, 9–20 (2012).
Perc, M., Gómez-Gardeñes, J., Szolnoki, A., Floría, L. M. & Moreno, Y. Evolutionary dynamics of group interactions on structured populations: a review. J. R. Soc. Interface 10, 20120997 (2013).
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).
Szolnoki, A., Perc, M. & Szabó, G. Topology-independent impact of noise on cooperation in spatial public goods games. Phys. Rev. E 80, 056109 (2009).
Trivers, R. L. The evolution of reciprocal altruism. Q. Rev. Biol. 46, 35–57 (1971).
Sigmund, K. Punish or perish? Retaliation and collaboration among humans. Trends Ecol. Evol. 22, 593–600 (2007).
Nowak, M. A. & Sigmund, K. Evolution of indirect reciprocity by image scoring. Nature 393, 573–577 (1998).
Milinski, M., Semmann, D., Bakker, T. C. M. & Krambeck, H.-J. Cooperation through indirect reciprocity: image scoring or standing strategy? Proc. R. Soc. Lond. B 268, 2495–2501 (2001).
Nax, H. H., Perc, M., Szolnoki, A. & Helbing, D. Stability of cooperation under image scoring in group interactions. Sci. Rep. 5, 12145 (2015).
Fehr, E. Donat lose your reputation. Nature 432, 449–450 (2004).
Gächter, S. Reputation and reciprocity: Consequences for the labour relation. Scand. J. Econ. 104, 1–26 (2002).
Fu, F., Hauert, C., Nowak, M. A. & Wang, L. Reputation-based partner choice promotes cooperation in social networks. Phys. Rev. E 78, 026117 (2008).
Latora, V, Nicosia, V. & Russo, G. Complex Networks: Principles, Methods and Applications (Cambridge Univ. Press, 2017).
Berge, C. Hypergraphs: Combinatorics of Finite Sets (Elsevier, 1984).
Battiston, F. et al. Networks beyond pairwise interactions: structure and dynamics. Phys. Rep. 874, 1–92 (2020).
Gómez-Gardeñes, J., Romance, M., Criado, R., Vilone, D. & Sánchez, A. Evolutionary games defined at the network mesoscale: the public goods game. Chaos 21, 016113 (2011).
Gómez-Gardeñes, J., Vilone, D. & Sánchez, A. Disentangling social and group heterogeneities: public goods games on complex networks. Europhys. Lett. 95, 68003 (2011).
Peña, J. & Rochat, Y. Bipartite graphs as models of population structures in evolutionary multiplayer games. PLoS ONE 7, e44514 (2012).
Wu, J. & Zusai, D. A potential game approach to modelling evolution in a connected society. Nat. Hum. Behav. 3, 604–610 (2019).
Taylor, P. & Jonker, L. Evolutionary stable strategies and game dynamics. Math. Biosci. 40, 145–156 (1978).
Wu, L., Wang, D. & Evans, J. A. Large teams develop and small teams disrupt science and technology. Nature 566, 378–382 (2019).
Wuchty, S., Jones, B. F. & Uzzi, B. The increasing dominance of teams in production of knowledge. Science 316, 1036–1039 (2007).
Klug, M. & Bagrow, J. P. Understanding the group dynamics and success of teams. R. Soc. Open Sci. 3, 160007 (2016).
Bettencourt, L. M. A., Lobo, J., Helbing, D., Kühnert, C. & West, G. B. Growth, innovation, scaling, and the pace of life in cities. Proc. Natl Acad. Sci. USA 104, 7301–7306 (2007).
Bettencourt, L. M. A. The origins of scaling in cities. Science 340, 1438 (2013).
Battiston, F. et al. Taking census of physics. Nat. Rev. Phys. 1, 89–97 (2019).
Milojević, S. Principles of scientific research team formation and evolution. Proc. Natl Acad. Sci. USA 111, 3984–3989 (2014).
Wang, Z., Wang, L., Szolnoki, A. & Perc, M. Evolutionary games on multilayer networks: a colloquium. Eur. Phys. J. B 88, 124 (2015).
Rand, D. G., Dreber, A., Ellingsen, T., Fudenberg, D. & Nowak, M. A. Positive interactions promote public cooperation. Science 325, 1272–1275 (2009).
Andreoni, J., Harbaugh, W. & Vesterlund, L. The carrot or the stick: rewards, punishments, and cooperation. Am. Econ. Rev. 93, 893–902 (2003).
Gächter, S., Renner, E. & Sefton, M. The long-run benefits of punishment. Science 322, 1510 (2008).
Boyd, R., Gintis, H. & Bowles, S. Coordinated punishment of defectors sustains cooperation and can proliferate when rare. Science 328, 617–620 (2010).
Jordan, J. J., Hoffman, M., Bloom, P. & Rand, D. G. Third-party punishment as a costly signal of trustworthiness. Nature 530, 473–476 (2016).
Eliad-Badt, E. Decomposition of the complete hypergraph into stars. Discrete Math. 71, 107–117 (1988).
Jhun, B., Minjae, J. & Kahng, B. Simplicial SIS model in scale-free uniform hypergraph. J. Stat. Mech. https://doi.org/10.1088/1742-5468/ab5367 (2019).
APS Dataset https://journals.aps.org/datasets.
Acknowledgements
U.A.-R. acknowledges support from the Spanish Government through the Maria de Maeztu excellence accreditation 2018-2022 (ref. MDM-2017-0714) and from the Basque Government through the postdoctoral programme (ref. POS-2017-1-0022). F.B. acknowledges partial support from ERC synergy grant 810115 (DYNASNET). V.L. acknowledges support from the Leverhulme Trust Research Fellowship ‘CREATE: the network components of creativity and success’. Y.M. acknowledges partial support from the Government of Aragón and FEDER funds, Spain through grant E36-20R to FENOL, by MINECO and FEDER funds (grant FIS2017-87519-P) and from Intesa Sanpaolo Innovation Center. M.P. was supported by the Slovenian Research Agency (grant numbers J1-2457, J1-9112 and P1-0403). G.F.A. acknowledges support from Intesa Sanpaolo Innovation Center. We thank M. Clarin from COSNET Lab for help and assistance with the figures. The funders had no role in study design, data collection and analysis, decision to publish or preparation of the manuscript.
Author information
Authors and Affiliations
Contributions
U.A.-R., F.B. and V.L. conceived the study with contributions from G.F.A., M.P. and Y.M. U.A.-R. performed the calculations. U.A.-R., F.B., G.F.A., M.P., Y.M. and V.L. analysed the data and discussed the results. U.A.-R., F.B., G.F.A., M.P., Y.M. and V.L. wrote the paper.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Peer review information Primary Handling Editor: Aisha Bradshaw
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Supplementary Information
Supplementary Discussion, Supplementary Figs. 1–5 and Supplementary Table 1.
Rights and permissions
About this article
Cite this article
Alvarez-Rodriguez, U., Battiston, F., de Arruda, G.F. et al. Evolutionary dynamics of higher-order interactions in social networks. Nat Hum Behav 5, 586–595 (2021). https://doi.org/10.1038/s41562-020-01024-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/s41562-020-01024-1
This article is cited by
-
Collective dynamics of swarmalators with higher-order interactions
Communications Physics (2024)
-
Higher-order Granger reservoir computing: simultaneously achieving scalable complex structures inference and accurate dynamics prediction
Nature Communications (2024)
-
Strategy evolution on higher-order networks
Nature Computational Science (2024)
-
Effects of high-order interactions on synchronization of a fractional-order neural system
Cognitive Neurodynamics (2024)
-
Higher-order organization of multivariate time series
Nature Physics (2023)