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Strategy evolution on dynamic networks

Nature Computational Science volume 3, pages 763–776 (2023)Cite this article

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Abstract

Models of strategy evolution on static networks help us understand how population structure can promote the spread of traits like cooperation. One key mechanism is the formation of altruistic spatial clusters, where neighbors of a cooperative individual are likely to reciprocate, which protects prosocial traits from exploitation. However, most real-world interactions are ephemeral and subject to exogenous restructuring, so that social networks change over time. Strategic behavior on dynamic networks is difficult to study, and much less is known about the resulting evolutionary dynamics. Here we provide an analytical treatment of cooperation on dynamic networks, allowing for arbitrary spatial and temporal heterogeneity. We show that transitions among a large class of network structures can favor the spread of cooperation, even if each individual social network would inhibit cooperation when static. Furthermore, we show that spatial heterogeneity tends to inhibit cooperation, whereas temporal heterogeneity tends to promote it. Dynamic networks can have profound effects on the evolution of prosocial traits, even when individuals have no agency over network structures.

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Fig. 1: Evolutionary games on dynamic networks.
Fig. 2: Transitions between networks that contain dense and sparse communities.
Fig. 3: Dynamic structures facilitate cooperation for a broad range of population sizes and network transition rates.
Fig. 4: Intuition for how dynamic structures can facilitate cooperation.
Fig. 5: Evolution of cooperation on diverse dynamic structures.
Fig. 6: Effects of spatial and temporal burstiness on cooperation.

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Data availability

Source data are provided with this paper.

Code availability

All numerical calculations and computational simulations were performed in Julia 1.4.1. All data analyses were performed in Matlab R2020a. All codes have been deposited into the publicly available GitHub repository at https://github.com/qisu1991/DynamicNetworks ref. 59.

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Acknowledgements

This work is supported by the Simons Foundation (Math+X Grant to the University of Pennsylvania). J.B.P. acknowledges generous support by the John Templeton Foundation (grant no. 62281) and the David & Lucille Packard Foundation.

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Author notes

  1. These authors contributed equally: Qi Su, Alex McAvoy.

Authors and Affiliations

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

    Qi Su

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

    Qi Su

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

    Qi Su

  4. School of Data Science and Society, University of North Carolina at Chapel Hill, Chapel Hill, NC, USA

    Alex McAvoy

  5. Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC, USA

    Alex McAvoy

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

    Joshua B. Plotkin

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

    Joshua B. Plotkin

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Contributions

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

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Correspondence to Qi Su or Alex McAvoy.

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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: Fernando Chirigati, in collaboration with the Nature Computational Science team. Peer reviewer reports are available.

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Extended data

Extended Data Fig. 1 The cooperation-promoting effects of structure transitions as the sizes of the two communities vary.

The dynamic network is illustrated in Fig. 2a, with a fraction a (resp. 1 − a) of nodes in the top (resp. bottom) community. The critical benefit-to-cost ratio, (b/c)*, is shown as a function of a. The dots are the results of numerical calculations with N = 10, 000 and the lines are analytical approximations for sufficiently large N. The rescaled duration is t = 1.

Source data

Extended Data Fig. 2 A qualitative analysis of cooperator expansion in the star community. and cooperator reduction in the complete community.

a, We consider a configuration where the hub node and n − 1 other nodes are cooperators in each community, and the rest are defectors. The fecundity of the hub node in the star community is given by \({F}_{{{{\rm{star}}}},{{{\rm{hub}}}}}=\exp [\delta (nb-Nc/2)]\). And the fecundity of a cooperator the other nodes is given by \({F}_{{{{\rm{star}}}},{{{\rm{C}}}}}=\exp [\delta (b-c)]\); whereas the fecundity of a defector node is given by \({F}_{{{{\rm{star}}}},{{{\rm{D}}}}}=\exp [\delta b]\). We can also derive expressions for fecundities in the complete community: \({F}_{{{{\rm{comp}}}},{{{\rm{hub}}}}}=\exp [\delta (nb-Nc/2)]\), \({F}_{{{{\rm{comp}}}},{{{\rm{C}}}}}=\exp [\delta ((n-1)b-(N/2-1)c)]\), \({F}_{{{{\rm{comp}}}},{{{\rm{D}}}}}=\exp [\delta nb]\). Finally, we can calculate the expected change in the number of cooperators in each community, denoted by Δstar in the star community and Δcomp in the complete community. b, The expected change in the number of cooperators in the star community and in the complete community, for across the full range sub-graph sizes, n. Dots are obtained by Δstar and Δcomp. Horizontal lines mark the average change across all possibilities of n. The increase in the number of cooperators in the star community exceeds the decrease in the number of cooperators in the complete community. Parameters: N = 40, δ = 0.1, b = 10, c = 1.

Source data

Extended Data Fig. 3 Cooperation-promoting effects of dynamic multi-community networks.

We consider networks made up of eight communities connected via hub nodes (see Fig. 5a; panels a and b here) and via leaf nodes (see Fig. 5b; panels c and d here). a, c, The critical ratio (b/c)* as a function of population size N, for the rescaled duration t = 1. b, d, The critical ratio (b/c)* as a function of the rescaled duration t, for N = 200.

Source data

Extended Data Fig. 4 Cooperation-promoting effects of structure transitions among more than two networks, and when networks differ in a small fraction of connections.

a, Structure transitions among three networks. Every network transitions to another network with probability 1/(2tN) and remains unchanged otherwise. b, Structure transitions between multi-community networks in which the two networks differ in only two communities. We take N = 150 in a and N = 64 in b, and the rescaled duration is t = 1.

Extended Data Fig. 5 Dynamic networks promote and accelerate the fixation of cooperators.

We consider the network with a star community and a complete-graph community with N = 16 and a = 0.5 (see Fig. 2a). a, Fixation probability of cooperators as a function of the rescaled duration, t, in network 1 and in the dynamic network. The dynamic network leads to the larger fixation probability of cooperators than in network 1. b, Conditional and unconditional fixation times as functions of the rescaled duration, t. Both the conditional and unconditional times in the dynamic networks are smaller than in network 1. We take selection intensity δ = 0.1.

Source data

Extended Data Fig. 6 Dynamic structures can inhibit spite for a broad range of population sizes, under birth-death updating.

We consider transitions between the two networks shown in Fig. 2a in the main text, each composed of a sparse community and a dense community. The figure plots the critical benefit-to-cost ratio for cooperation as a function of population size, N, for a = 0.5 and t = 1. Under birth-death updating, spite rather than cooperation can evolve, as indicated by a negative critical ratio (b/c)*. These dynamic networks, compared with static networks, reduce the magnitude of the critical ratio, thus making spiteful behavior harder to evolve.

Source data

Extended Data Fig. 7 Dynamic structures can facilitate cooperation under strong selection.

We consider transitions between the two networks shown in Fig. 2a in the main text, each composed of a sparse community and a dense community. The figure plots the fixation probability of cooperation versus defection, ρC − ρD, as a function of the benefit b in the donation game, for N = 12, a = 0.5, and t = 1. These dynamic networks are more beneficial than static networks, for the evolution of cooperation. Selection strength: δ = 0.1.

Source data

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Su, Q., McAvoy, A. & Plotkin, J.B. Strategy evolution on dynamic networks. Nat Comput Sci 3, 763–776 (2023). https://doi.org/10.1038/s43588-023-00509-z

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