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Evolutionary dynamics of higher-order interactions in social networks



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.

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Fig. 1: Higher-order versus pairwise interactions in a PGG.
Fig. 2: A PGG with higher-order interactions in URHs.
Fig. 3: A PGG with higher-order interactions in hyperdegree-heterogeneous random hypergraphs.
Fig. 4: A PGG with higher-order interactions in order-heterogeneous random hypergraphs.
Fig. 5: Synergy factors of scientific collaborations.

Data availability

The APS dataset is provided by the APS at:

Code availability

Custom code that supports the findings of this study is available from the corresponding author upon request.


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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.

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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.

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Correspondence to Unai Alvarez-Rodriguez.

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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).

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