Evolution of cooperation on dynamical graphs
Open accessAbstract
There are two key characteristic of animal and human societies: (1) degree heterogeneity, meaning that not all individual have the same number of associates; and (2) the interaction topology is not static, i.e. either individuals interact with different set of individuals at different times of their life, or at least they have different associations than their parents. Earlier works have shown that population structure is one of the mechanisms promoting cooperation. However, most studies had assumed that the interaction network can be described by a regular graph (homogeneous degree distribution). Recently there are an increasing number of studies employing degree heterogeneous graphs to model interaction topology. But mostly the interaction topology was assumed to be static. Here we investigate the fixation probability of the cooperator strategy in the prisoner's dilemma, when interaction network is a random regular graph, a random graph or a scale-free graph and the interaction network is allowed to change. We show that the fixation probability of the cooperator strategy is lower when the interaction topology is described by a dynamical graph compared to a static graph. Even a limited network dynamics significantly decreases the fixation probability of cooperation, an effect that is mitigated stronger by degree heterogeneous networks topology than by a degree homogeneous one. We have also found that from the considered graph topologies the decrease of fixation probabilities due to graph dynamics is the lowest on scale-free graphs.
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
Switching between two random networks tends to impede cooperation relative to each separate network.
“For example, switching between two random networks tends to impede cooperation relative to each separate network [12].”
From Strategy evolution on dynamic networks (2023) · cites this paper as [12]
The Results section reports that a cooperator’s fixation probability is lower on a changing graph than on the corresponding static graph, including random and random-regular graphs. The Methods section describes random rewiring after updates, not switching between two separate random networks; that specific comparison is unsupported. (LLM, conf=0.88)
Experimental — accuracy not yet measuredThe result that spatial burstiness inhibits cooperation aligns with previous simulation studies of random network rewiring.
“This result aligns with previous simulation studies of random network rewiring [12].”
From Strategy evolution on dynamic networks (2023) · cites this paper as [12]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Cites
Statistical mechanics of complex networks
Albert, R., & Barabási, A.-L. (2002) · Reviews of Modern Physics · doi:10.1103/revmodphys.74.47 · Cited by 20,471 · see citing works
Five Rules for the Evolution of Cooperation
Nowak, M. A. (2006) · Science · doi:10.1126/science.1133755 · Cited by 5,894 · see citing works
Specificity and Stability in Topology of Protein Networks
Maslov, S., & Sneppen, K. (2002) · Science · doi:10.1126/science.1065103 · Cited by 3,051 · see citing works
A simple rule for the evolution of cooperation on graphs and social networks
Ohtsuki, H., Hauert, C., Lieberman, E., & Nowak, M. A. (2006) · Nature · doi:10.1038/nature04605 · Cited by 2,208 · see citing works
Scale-Free Networks Provide a Unifying Framework for the Emergence of Cooperation
Santos, F. C., & Pacheco, J. M. (2005) · Physical Review Letters · doi:10.1103/physrevlett.95.098104 · Cited by 1,824 · see citing works
Evolutionary dynamics of social dilemmas in structured heterogeneous populations
Santos, F. C., Pacheco, J. M., & Lenaerts, T. (2006) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.0508201103 · Cited by 1,009 · see citing works
Scale-free topology of e-mail networks
Ebel, H., Mielsch, L.-I., & Bornholdt, S. (2002) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.66.035103 · Cited by 945 · see citing works
The emergent properties of a dolphin social network
Lusseau, D. (2003) · Proceedings of the Royal Society of London. Series B: Biological Sciences · doi:10.1098/rsbl.2003.0057 · Cited by 844 · see citing works
Cooperation among animals: an evolutionary perspective
(1997) · Choice Reviews Online · doi:10.5860/choice.35-0904 · Cited by 709 · see citing works
Coevolution of Strategy and Structure in Complex Networks with Dynamical Linking
Pacheco, J. M., Traulsen, A., & Nowak, M. A. (2006) · Physical Review Letters · doi:10.1103/physrevlett.97.258103 · Cited by 697 · see citing works
Matriarchs As Repositories of Social Knowledge in African Elephants
McComb, K., Moss, C., Durant, S. M., Baker, L., & Sayialel, S. (2001) · Science · doi:10.1126/science.1057895 · Cited by 651 · see citing works
The socioecology of elephants: analysis of the processes creating multitiered social structures
Wittemyer, G., Douglas-Hamilton, I., & Getz, W. M. (2005) · Animal Behaviour · doi:10.1016/j.anbehav.2004.08.018 · Cited by 591 · see citing works
Cooperation Prevails When Individuals Adjust Their Social Ties
Santos, F. C., Pacheco, J. M., & Lenaerts, T. (2006) · PLoS Computational Biology · doi:10.1371/journal.pcbi.0020140 · Cited by 556 · see citing works
The replicator equation on graphs
Ohtsuki, H., & Nowak, M. A. (2006) · Journal of Theoretical Biology · doi:10.1016/j.jtbi.2006.06.004 · Cited by 459 · see citing works
Evolution of cooperation in a finite homogeneous graph
Taylor, P. D., Day, T., & Wild, G. (2007) · Nature · doi:10.1038/nature05784 · Cited by 302 · see citing works
Towards effective payoffs in the prisoner’s dilemma game on scale-free networks
Szolnoki, A., Perc, M., & Danku, Z. (2007) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2007.11.021 · Cited by 298 · see citing works
Sexual selection in primates: New and comparative perspectives
Kappeler, P. M. (2002) · Evolutionary Anthropology Issues News and Reviews · doi:10.1002/evan.10023 · Cited by 293 · see citing works
Evolutionary prisoner’s dilemma games with voluntary participation
Szabó, G., & Hauert, C. (2002) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.66.062903 · Cited by 287 · see citing works
Sundaresan, S. R., Fischhoff, I. R., Dushoff, J. G., & Rubenstein, D. I. (2006) · Oecologia · doi:10.1007/s00442-006-0553-6 · Cited by 271 · see citing works
Active linking in evolutionary games
Pacheco, J. M., Traulsen, A., & Nowak, M. A. (2006) · Journal of Theoretical Biology · doi:10.1016/j.jtbi.2006.06.027 · Cited by 259 · see citing works
Analysing animal social structure
Whitehead, H. (1997) · Animal Behaviour · doi:10.1006/anbe.1996.0358 · Cited by 233 · see citing works
Lehmann, J., & Boesch, C. (2004) · Behavioral Ecology and Sociobiology · doi:10.1007/s00265-004-0781-x · Cited by 218 · see citing works
Superalliance of bottlenose dolphins
Connor, R. C., Heithaus, M. R., & Barré, L. M. (1999) · Nature · doi:10.1038/17501 · Cited by 215 · see citing works
Recent developments in the study of wild chimpanzee behavior
Mitani, J. C., Watts, D. P., & Muller, M. N. (2002) · Evolutionary Anthropology Issues News and Reviews · doi:10.1002/evan.10008 · Cited by 213 · see citing works
Fission–fusion social systems as a strategy for coping with ecological constraints: a primate case
Lehmann, J., Korstjens, A. H., & Dunbar, R. (2006) · Evolutionary Ecology · doi:10.1007/s10682-006-9141-9 · Cited by 210 · see citing works
Investigating structure and temporal scale in social organizations using identified individuals
Whitehead, H. (1995) · Behavioral Ecology · doi:10.1093/beheco/6.2.199 · Cited by 206 · see citing works
Evolutionary graph theory: Breaking the symmetry between interaction and replacement
Ohtsuki, H., Pacheco, J. M., & Nowak, M. A. (2007) · Journal of Theoretical Biology · doi:10.1016/j.jtbi.2007.01.024 · Cited by 191 · see citing works
Spinner dolphins in a remote Hawaiian atoll: social grouping and population structure
Karczmarski, L., Würsig, B., Gailey, G., Larson, K., & VANDERLIP, C. A. (2005) · Behavioral Ecology · doi:10.1093/beheco/ari028 · Cited by 181 · see citing works
Behavioral Ecology: Social Organization in Fission–Fusion Societies
Couzin, I. D. (2006) · Current Biology · doi:10.1016/j.cub.2006.02.042 · Cited by 179 · see citing works
Ramos‐Fernández, G. (2005) · International Journal of Primatology · doi:10.1007/s10764-005-6459-z · Cited by 156 · see citing works
Rubenstein, D. I., & Hack, M. A. (2004) · Cambridge University Press eBooks · doi:10.1017/cbo9780511542459.017 · Cited by 149 · see citing works
Behavioural evidence for social units in long-finned pilot whales
Ottensmeyer, C. A., & Whitehead, H. (2003) · Canadian Journal of Zoology · doi:10.1139/z03-127 · Cited by 144 · see citing works
Rank-related partner choice in the fission–fusion society of the spotted hyena (Crocuta crocuta)
Smith, J. E., Memenis, S. K., & Holekamp, K. E. (2006) · Behavioral Ecology and Sociobiology · doi:10.1007/s00265-006-0305-y · Cited by 144 · see citing works
Cooperation for volunteering and partially random partnerships
Szabó, G., & Vukov, J. (2004) · Physical Review E · doi:10.1103/physreve.69.036107 · Cited by 138 · see citing works
Evolutionary stability on graphs
Ohtsuki, H., & Nowak, M. A. (2008) · Journal of Theoretical Biology · doi:10.1016/j.jtbi.2008.01.005 · Cited by 124 · see citing works
Disentangling association patterns in fission–fusion societies using African buffalo as an example
Cross, P. C., Lloyd‐Smith, J. O., & Getz, W. M. (2004) · Animal Behaviour · doi:10.1016/j.anbehav.2004.08.006 · Cited by 121 · see citing works
The evolution of helping and harming on graphs: the return of the inclusive fitness effect
Lehmann, L., Keller, L., & Sumpter, D. J. T. (2007) · Journal of Evolutionary Biology · doi:10.1111/j.1420-9101.2007.01414.x · Cited by 97 · see citing works
Nowak, M. A., & Sigmund, K. (2000) · Cambridge University Press eBooks · doi:10.1017/cbo9780511525537.010 · Cited by 89 · see citing works
Evolutionary game dynamics with non-uniform interaction rates
Taylor, C., & Nowak, M. A. (2006) · Theoretical Population Biology · doi:10.1016/j.tpb.2005.06.009 · Cited by 84 · see citing works
Do pigs form preferential associations?
Durrell, J. L., Sneddon, I., O’Connell, N. E., & Whitehead, H. (2004) · Applied Animal Behaviour Science · doi:10.1016/j.applanim.2004.05.003 · Cited by 79 · see citing works
Instant Messaging as a Scale-Free NetworkPreprint
Smith, R. D. (2002) · arXiv (Cornell University) · doi:10.48550/arxiv.cond-mat/0206378 · Cited by 59 · see citing works
Equilibrium Statistical Mechanicsof Network Structures
Farkas, I. J., Derényi, I., Palla, G., & Vicsek, T. (2004) · Lecture notes in physics · doi:10.1007/978-3-540-44485-5_8 · Cited by 26 · see citing works
Correlations in scale-free networks: Tomography and percolation
Xulvi-Brunet, R., Pietsch, W., & Sokolov, I. M. (2003) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.68.036119 · Cited by 24 · see citing works
Flagged first, then by citations
2 of the 46 references couldn’t be matched to a known paper.
Research works printed but not linked: 1
- Barabási, A.-L., 2002. Linked. Perseus Publishing, Cambridge, MA.
Cited by
Sorted by citations
Coevolutionary games—A mini review
Perc, M., & Szolnoki, A. (2009) · Biosystems · doi:10.1016/j.biosystems.2009.10.003 · Cited by 1,939 · see citing works
Evolutionary forces favoring intragroup coalitions among spotted hyenas and other animals
Smith, J. E., HORN, R. C. V., Powning, K. S., Cole, A. R., Graham, K. E., Memenis, S. K., & Holekamp, K. E. (2010) · Behavioral Ecology · doi:10.1093/beheco/arp181 · Cited by 218 · see citing works
Strategy evolution on dynamic networks
Su, Q., McAvoy, A., & Plotkin, J. B. (2023) · Nature Computational Science · doi:10.1038/s43588-023-00509-z · Cited by 88 · see citing works
“For example, switching between two random networks tends to impede cooperation relative to each separate network 12 .”
BackgroundResultDifferent Reactions to Adverse Neighborhoods in Games of Cooperation
Zhang, C., Zhang, J., Weissing, F. J., Perc, M., Xie, G., & Wang, L. (2012) · PLoS ONE · doi:10.1371/journal.pone.0035183 · Cited by 46 · see citing works
Ecological Determinants of Social Systems
Chaverri, G., & Kunz, T. (2010) · Advances in the study of behavior · doi:10.1016/s0065-3454(10)42009-4 · Cited by 40 · see citing works
Coggan, H., & Page, K. M. (2022) · Journal of The Royal Society Interface · doi:10.1098/rsif.2022.0346 · Cited by 30 · see citing works
Dynamic social networks facilitate cooperation in the N -player Prisoner’s Dilemma
Rezaei, G., & Kirley, M. (2012) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2012.06.071 · Cited by 29 · see citing works
Adaptive-network models of collective dynamics
Zschaler, G. (2012) · The European Physical Journal Special Topics · doi:10.1140/epjst/e2012-01648-5 · Cited by 28 · see citing works
The ‘Hawk-Dove’ Game and the Speed of the Evolutionary Process in Small Heterogeneous Populations
Voelkl, B. (2010) · Games · doi:10.3390/g1020103 · Cited by 20 · see citing works
Modularity in protein structures: study on all-alpha proteins
Khan, T., & Ghosh, I. (2015) · Journal of Biomolecular Structure and Dynamics · doi:10.1080/07391102.2014.1003969 · Cited by 20 · see citing works
Stirring does not make populations well mixed
Herrerías‐Azcué, F., Pérez-Muñuzuri, V., & Galla, T. (2018) · Scientific Reports · doi:10.1038/s41598-018-22062-w · Cited by 20 · see citing works
Evolutionary Game Model With Group Decision-Making in Signed Social Networks
Qu, C., Ji, C., & Zhang, M. (2022) · IEEE Transactions on Computational Social Systems · doi:10.1109/tcss.2022.3209877 · Cited by 17 · see citing works
Double-dealing behavior potentially promotes cooperation in evolutionary prisoner's dilemma games
Dai, Q., Li, H., Cheng, H., Li, Y., & Yang, J. (2010) · New Journal of Physics · doi:10.1088/1367-2630/12/11/113015 · Cited by 16 · see citing works
Survivor’s dilemma: Defend the group or flee?
Garay, J., & Varga, Z. (2011) · Theoretical Population Biology · doi:10.1016/j.tpb.2011.08.003 · Cited by 12 · see citing works
Optional contributions have positive effects for volunteering public goods games
Song, Q.-Q., Li, Z., Fu, C.-H., & Wang, L.-S. (2011) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2011.07.025 · Cited by 11 · see citing works
Evolving Cooperation in the N-player Prisoner’s Dilemma: A Social Network Model
Rezaei, G., Kirley, M., & Pfau, J. (2009) · Lecture notes in computer science · doi:10.1007/978-3-642-10427-5_5 · Cited by 10 · see citing works
The impact of dynamic linking on cooperation on complex networks
Zhao, X., Hu, K., Tao, Y., Jin, L., & Ши, Л. (2024) · Chaos An Interdisciplinary Journal of Nonlinear Science · doi:10.1063/5.0221942 · Cited by 9 · see citing works
Public Goods Games in Disease Evolution and Spread
Morison, C., Fic, M., Marcou, T., Mohamadichamgavi, J., Antón, J., Sayyar, G., Stein, A., Bastian, F., Krakovská, H., Krishnan, N., et al. (2025) · Dynamic Games and Applications · doi:10.1007/s13235-025-00619-5 · Cited by 8 · see citing works
The impact of truncation selection and diffusion on cooperation in spatial games
Morsky, B., & Bauch, C. T. (2019) · Journal of Theoretical Biology · doi:10.1016/j.jtbi.2019.01.023 · Cited by 8 · see citing works
Kun, Á., Boza, G., & Scheuring, I. (2010) · BMC Evolutionary Biology · doi:10.1186/1471-2148-10-173 · Cited by 8 · see citing works
Self-citationCrossover between structured and well-mixed networks in an evolutionary prisoner’s dilemma game
Dai, Q., Cheng, H., Li, H., Li, Y., Zhang, M., & Yang, J. (2011) · Physical Review E · doi:10.1103/physreve.84.011103 · Cited by 8 · see citing works
Graph Transformations and Game Theory: A Generative Mechanism for Network Formation
Cavaliere, M., Csikász‐Nagy, A., & Jordán, F. (2008) · Unitn Eprints Research (Università Degli Studi di Trento) · Cited by 6 · see citing works
A Parrondo paradox in susceptible-infectious-susceptible dynamics over periodic temporal networks
Sejunti, M. I., Taylor, D., & Masuda, N. (2024) · Mathematical Biosciences · doi:10.1016/j.mbs.2024.109336 · Cited by 5 · see citing works
Montoreano, C., & Jaffé, K. (2013) · arXiv (Cornell University) · doi:10.48550/arxiv.1311.4407 · Cited by 4 · see citing works
Evolution of mixed strategies for social dilemmas on structured networks
Yazicioglu, A. Y., Ma, X., & Altunbasak, Y. (2011) · IEEE International Conference on Networking, Sensing and Control (ICNSC) · doi:10.1109/icnsc.2011.5874951 · Cited by 2 · see citing works
Liu, P., & Liu, J. (2017) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2017.03.004 · Cited by 1 · see citing works
Evolutionary models in structured populations
Hadjichrysanthou, C. (2012) · UPT. Syiah Kuala University Library (Syiah Kuala University) · Cited by 1 · see citing works
Light weight security architecture for ad hoc networks
Joseph, J. F. C. (2010) · doi:10.32657/10356/41881
Flipping the intuition for games on dynamic networks
Allen, B. (2023) · Nature Computational Science · doi:10.1038/s43588-023-00513-3
- Authors
- Kun, Á., & Scheuring, I.
- Affiliations
- Eötvös Loránd University (HU)
- Corresponding author
- Kun, Ádám
- Venue
- Biosystems
- Publisher
- Elsevier BV
- Published
- 2009-04
- Volume
- 96
- Issue
- 1
- Pages
- 65-68
- ISSN
- 0303-2647 (ISSN-L)
- OpenAlex
- W2149500634
- Cited by
- 29
- References
- 46
- Citations by year
- 2025: 1 · 2024: 2 · 2023: 2 · 2022: 2 · 2019: 1 · 2018: 1 · 2017: 1 · 2015: 1 · 2013: 1 · 2012: 4
- FWCI
- 14.99
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Biosystems (publishedVersion)
- PubMed (publishedVersion)
- IIASA PURE (International Institute of Applied Systems Analysis) (acceptedVersion)
- Repository of the Academy's Library (Library of the Hungarian Academy of Sciences) (acceptedVersion)
- IIASA PURE (International Institute of Applied Systems Analysis) (submittedVersion)
- Funders
- Hungarian Scientific Research Fund (T049692)
- Topics
- Social Sciences › Sociology and Political Science › Evolutionary Game Theory and Cooperation
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Complex Network Analysis Techniques
- Health Sciences › Medicine › Public Health, Environmental and Occupational Health › Mathematical and Theoretical Epidemiology and Ecology Models
- Keywords
- evolution of cooperation, fixation probability, prisoner's dilemma, degree heterogeneity, scale-free networks, random regular graphs, random graphs, network dynamics, population structure
- MeSH terms
- Algorithms, Animals, Computer Simulation, Cooperative Behavior, Biological Evolution, Game Theory, Genetics, Population, Humans, Models, Genetic
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
