{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T13:57:51Z","timestamp":1790863071136,"version":"4.1.0"},"reference-count":51,"publisher":"AIP Publishing","issue":"10","content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2026,10,1]]},"abstract":"<jats:p>Cooperation is crucial for all levels of living systems. The last two decades have seen progress in revealing the mechanisms for cooperation in structured populations. Most of those analytical works have an explicit assumption, i.e., weak rationality or high temperature in statistical physical terms. It is still far from known how rationality alters the fate of cooperation, although simulation results have provided some intuitions. To be specific, the analytical insight is still lacking under what conditions cooperation can still be promoted, given that it cannot be favored by weak rationality. To this end, we investigate a coevolutionary model where both strategy and network evolve simultaneously. We give a sufficient condition under which cooperation cannot be favored for weak-selection limit, but it can be rescued by enhancing rationality alone. Furthermore, we give the critical rationality analytically. Counterintuitively, we show that the critical rationality can be larger if cooperator\u2013defector links are more fragile, indicating that robust cooperator\u2013defector social relationships can rescue cooperation. We, thus, show that increasing rationality intensity alone can rescue cooperation, where it cannot be present under weak-selection limit. Our work calls for an extrapolation from weak rationality in evolutionary game theory.<\/jats:p>","DOI":"10.1063\/5.0354201","type":"journal-article","created":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T13:05:44Z","timestamp":1790859944000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["A novel route for saving cooperation on dynamical networks: Enhancing rationality"],"prefix":"10.1063","volume":"36","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9665-7984","authenticated-orcid":false,"given":"Yuanxin","family":"Liu","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Beijing University of Posts and Telecommunications , Beijing 100876,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5787-9536","authenticated-orcid":false,"given":"Bin","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Beijing University of Posts and Telecommunications , Beijing 100876,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2026,10,1]]},"reference":[{"key":"2026100108565944300_c1","doi-asserted-by":"publisher","first-page":"1560","DOI":"10.1126\/science.1133755","article-title":"Five rules for the evolution of cooperation","volume":"314","year":"2006","journal-title":"Science"},{"key":"2026100108565944300_c2","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1016\/j.physrep.2007.04.004","article-title":"Evolutionary games on graphs","volume":"446","year":"2007","journal-title":"Phys. Rep."},{"key":"2026100108565944300_c3","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.physrep.2017.05.004","article-title":"Statistical physics of human cooperation","volume":"687","year":"2017","journal-title":"Phys. Rep."},{"key":"2026100108565944300_c4","doi-asserted-by":"publisher","first-page":"109","DOI":"10.1016\/j.biosystems.2009.10.003","article-title":"Coevolutionary games\u2014A mini review","volume":"99","year":"2010","journal-title":"BioSystems"},{"key":"2026100108565944300_c5","doi-asserted-by":"publisher","first-page":"093033","DOI":"10.1088\/1367-2630\/11\/9\/093033","article-title":"Emergence of multilevel selection in the prisoner\u2019s dilemma game on coevolving random networks","volume":"11","year":"2009","journal-title":"New J. Phys."},{"key":"2026100108565944300_c6","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1098\/rsif.2007.1229","article-title":"Adaptive coevolutionary networks: A review","volume":"5","year":"2008","journal-title":"J. R. Soc. Interface"},{"key":"2026100108565944300_c7","doi-asserted-by":"publisher","first-page":"e140","DOI":"10.1371\/journal.pcbi.0020140","article-title":"Cooperation prevails when individuals adjust their social ties","volume":"2","year":"2006","journal-title":"PLoS Comput. Biol."},{"key":"2026100108565944300_c8","doi-asserted-by":"publisher","first-page":"e26724","DOI":"10.1371\/journal.pone.0026724","article-title":"Evolution of interactions and cooperation in the spatial prisoner\u2019s dilemma game","volume":"6","year":"2011","journal-title":"PLoS One"},{"key":"2026100108565944300_c9","doi-asserted-by":"publisher","first-page":"258103","DOI":"10.1103\/PhysRevLett.97.258103","article-title":"Coevolution of strategy and structure in complex networks with dynamical linking","volume":"97","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"2026100108565944300_c10","doi-asserted-by":"publisher","first-page":"e11187","DOI":"10.1371\/journal.pone.0011187","article-title":"Evolution of cooperation on stochastic dynamical networks","volume":"5","year":"2010","journal-title":"PLoS One"},{"key":"2026100108565944300_c11","doi-asserted-by":"publisher","first-page":"109978","DOI":"10.1016\/j.jtbi.2019.08.009","article-title":"Vaccination dilemma on an evolving social network","volume":"483","year":"2019","journal-title":"J. Theor. Biol."},{"key":"2026100108565944300_c12","doi-asserted-by":"publisher","first-page":"111225","DOI":"10.1016\/j.jtbi.2022.111225","article-title":"Social relationship adjustments within the same sex promote marital bliss","volume":"550","year":"2022","journal-title":"J. Theor. Biol."},{"key":"2026100108565944300_c13","doi-asserted-by":"publisher","first-page":"522","DOI":"10.1016\/j.jtbi.2007.01.002","article-title":"Pairwise comparison and selection temperature in evolutionary game dynamics","volume":"246","year":"2007","journal-title":"J. Theor. Biol."},{"key":"2026100108565944300_c14","doi-asserted-by":"publisher","first-page":"e66560","DOI":"10.1371\/journal.pone.0066560","article-title":"Evolution of cooperation in a heterogeneous graph: Fixation probabilities under weak selection","volume":"8","year":"2013","journal-title":"PLoS One"},{"key":"2026100108565944300_c15","doi-asserted-by":"publisher","first-page":"103007","DOI":"10.1088\/1367-2630\/18\/10\/103007","article-title":"Interactive diversity promotes the evolution of cooperation in structured populations","volume":"18","year":"2016","journal-title":"New J. Phys."},{"key":"2026100108565944300_c16","doi-asserted-by":"publisher","first-page":"e1005059","DOI":"10.1371\/journal.pcbi.1005059","article-title":"Evolutionary games of multiplayer cooperation on graphs","volume":"12","year":"2016","journal-title":"PLoS Comput. Biol."},{"key":"2026100108565944300_c17","doi-asserted-by":"publisher","first-page":"25398","DOI":"10.1073\/pnas.1908936116","article-title":"Evolutionary dynamics with game transitions","volume":"116","year":"2019","journal-title":"Proc. Natl. Acad. Sci. U.S.A."},{"key":"2026100108565944300_c18","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1038\/nature21723","article-title":"Evolutionary dynamics on any population structure","volume":"544","year":"2017","journal-title":"Nature"},{"key":"2026100108565944300_c19","doi-asserted-by":"publisher","first-page":"2259","DOI":"10.1038\/s41467-020-16088-w","article-title":"Evolution of cooperation on temporal networks","volume":"11","year":"2020","journal-title":"Nat. Commun."},{"key":"2026100108565944300_c20","doi-asserted-by":"publisher","first-page":"763","DOI":"10.1038\/s43588-023-00509-z","article-title":"Strategy evolution on dynamic networks","volume":"3","year":"2023","journal-title":"Nat. Comput. Sci."},{"key":"2026100108565944300_c21","doi-asserted-by":"publisher","first-page":"338","DOI":"10.1038\/s41562-021-01241-2","article-title":"Evolution of prosocial behaviours in multilayer populations","volume":"6","year":"2022","journal-title":"Nat. Hum. Behav."},{"key":"2026100108565944300_c22","doi-asserted-by":"publisher","first-page":"e1011333","DOI":"10.1371\/journal.pcbi.1011333","article-title":"Evolutionary dynamics on sequential temporal networks","volume":"19","year":"2023","journal-title":"PLoS Comput. Biol."},{"key":"2026100108565944300_c23","doi-asserted-by":"publisher","first-page":"e1008695","DOI":"10.1371\/journal.pcbi.1008695","article-title":"Fixation probabilities in graph-structured populations under weak selection","volume":"17","year":"2021","journal-title":"PLoS Comput. Biol."},{"key":"2026100108565944300_c24","doi-asserted-by":"publisher","first-page":"046106","DOI":"10.1103\/PhysRevE.82.046106","article-title":"Universality of weak selection","volume":"82","year":"2010","journal-title":"Phys. Rev. E"},{"key":"2026100108565944300_c25","doi-asserted-by":"publisher","first-page":"e54639","DOI":"10.1371\/journal.pone.0054639","article-title":"Consolidating birth-death and death-birth processes in structured populations","volume":"8","year":"2013","journal-title":"PLoS One"},{"key":"2026100108565944300_c26","doi-asserted-by":"publisher","first-page":"e1003567","DOI":"10.1371\/journal.pcbi.1003567","article-title":"Evolutionary game dynamics in populations with heterogenous structures","volume":"10","year":"2014","journal-title":"PLoS Comput. Biol."},{"key":"2026100108565944300_c27","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1016\/j.jtbi.2013.12.016","article-title":"Gains from switching and evolutionary stability in multi-player matrix games","volume":"346","year":"2014","journal-title":"J. Theor. Biol."},{"key":"2026100108565944300_c28","doi-asserted-by":"publisher","first-page":"073035","DOI":"10.1088\/1367-2630\/14\/7\/073035","article-title":"How selection pressure changes the nature of social dilemmas in structured populations","volume":"14","year":"2012","journal-title":"New J. Phys."},{"key":"2026100108565944300_c29","doi-asserted-by":"publisher","first-page":"e1009611","DOI":"10.1371\/journal.pcbi.1009611","article-title":"Intriguing effects of selection intensity on the evolution of prosocial behaviors","volume":"17","year":"2021","journal-title":"PLoS Comput. Biol."},{"key":"2026100108565944300_c30","doi-asserted-by":"publisher","first-page":"022407","DOI":"10.1103\/PhysRevE.95.022407","article-title":"Evolutionary games on cycles with strong selection","volume":"95","year":"2017","journal-title":"Phys. Rev. E"},{"key":"2026100108565944300_c31","doi-asserted-by":"publisher","first-page":"109","DOI":"10.1007\/s00285-012-0622-x","article-title":"Measures of success in a class of evolutionary models with fixed population size and structure","volume":"68","year":"2014","journal-title":"J. Math. Biol."},{"key":"2026100108565944300_c32","doi-asserted-by":"publisher","first-page":"182","DOI":"10.3390\/g4020182","article-title":"Dynamic properties of evolutionary multi-player games in finite populations","volume":"4","year":"2013","journal-title":"Games"},{"key":"2026100108565944300_c33","doi-asserted-by":"publisher","first-page":"e1003381","DOI":"10.1371\/journal.pcbi.1003381","article-title":"Extrapolating weak selection in evolutionary games","volume":"9","year":"2013","journal-title":"PLoS Comput. Biol."},{"key":"2026100108565944300_c34","doi-asserted-by":"publisher","first-page":"013007","DOI":"10.1088\/1367-2630\/13\/1\/013007","article-title":"Selection pressure transforms the nature of social dilemmas in adaptive networks","volume":"13","year":"2011","journal-title":"New J. Phys."},{"key":"2026100108565944300_c35","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1016\/j.jtbi.2019.01.006","article-title":"Evolutionary game dynamics of the Wright-Fisher process with different selection intensities","volume":"465","year":"2019","journal-title":"J. Theor. Biol."},{"key":"2026100108565944300_c36","doi-asserted-by":"publisher","first-page":"062309","DOI":"10.1103\/PhysRevE.101.062309","article-title":"Strategy equilibrium in dilemma games with off-diagonal payoff perturbations","volume":"101","year":"2020","journal-title":"Phys. Rev. E"},{"key":"2026100108565944300_c37","doi-asserted-by":"publisher","first-page":"011909","DOI":"10.1103\/PhysRevE.74.011909","article-title":"Stochastic dynamics of invasion and fixation","volume":"74","year":"2006","journal-title":"Phys. Rev. E"},{"key":"2026100108565944300_c38","doi-asserted-by":"publisher","first-page":"469","DOI":"10.1038\/nature05784","article-title":"Evolution of cooperation in a finite homogeneous graph","volume":"447","year":"2007","journal-title":"Nature"},{"key":"2026100108565944300_c39","volume-title":"Bounded Rationality: The Adaptive Toolbox","year":"2002"},{"key":"2026100108565944300_c40","doi-asserted-by":"publisher","first-page":"502","DOI":"10.1038\/nature04605","article-title":"A simple rule for the evolution of cooperation on graphs and social networks","volume":"441","year":"2006","journal-title":"Nature"},{"key":"2026100108565944300_c41","doi-asserted-by":"publisher","first-page":"570","DOI":"10.1016\/j.jtbi.2009.03.035","article-title":"Strategy selection in structured populations","volume":"259","year":"2009","journal-title":"J. Theor. Biol."},{"key":"2026100108565944300_c42","doi-asserted-by":"publisher","first-page":"30007","DOI":"10.1209\/0295-5075\/86\/30007","article-title":"Resolving social dilemmas on evolving random networks","volume":"86","year":"2009","journal-title":"Europhys. Lett."},{"key":"2026100108565944300_c43","doi-asserted-by":"publisher","first-page":"112992","DOI":"10.1016\/j.automatica.2026.112992","article-title":"The stability of bi-polarization on dynamical directed graphs: An emergent game perspective","volume":"189","year":"2026","journal-title":"Automatica"},{"key":"2026100108565944300_c44","doi-asserted-by":"publisher","first-page":"20240547","DOI":"10.1098\/rsif.2024.0547","article-title":"Evolutionary mechanisms that promote cooperation may not promote social welfare","volume":"21","year":"2024","journal-title":"J. R. Soc. Interface"},{"key":"2026100108565944300_c45","first-page":"494","article-title":"Evolution of cooperation on different pairs of interaction and replacement networks with various intensity of selection","year":"2009"},{"key":"2026100108565944300_c46","doi-asserted-by":"publisher","first-page":"128959","DOI":"10.1016\/j.amc.2024.128959","article-title":"Diverse selection intensities resolve the cooperation dilemma induced by breaking the symmetry between interaction and learning","volume":"482","year":"2024","journal-title":"Appl. Math. Comput."},{"key":"2026100108565944300_c47","doi-asserted-by":"publisher","first-page":"312","DOI":"10.1038\/nature03204","article-title":"Evolutionary dynamics on graphs","volume":"433","year":"2005","journal-title":"Nature"},{"key":"2026100108565944300_c48","doi-asserted-by":"publisher","first-page":"108106","DOI":"10.1103\/PhysRevLett.98.108106","article-title":"Breaking the symmetry between interaction and replacement in evolutionary dynamics on graphs","volume":"98","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"2026100108565944300_c49","doi-asserted-by":"publisher","first-page":"20150881","DOI":"10.1098\/rsif.2015.0881","article-title":"Ordering structured populations in multiplayer cooperation games","volume":"13","year":"2016","journal-title":"J. R. Soc. Interface"},{"key":"2026100108565944300_c50","doi-asserted-by":"publisher","first-page":"15636","DOI":"10.1073\/pnas.1511366112","article-title":"Computational complexity of ecological and evolutionary spatial dynamics","volume":"112","year":"2015","journal-title":"Proc. Natl. Acad. Sci. U.S.A."},{"key":"2026100108565944300_c51","unstructured":"Y. Liu (2026). \u201cA novel route for saving cooperation on dynamical networks,\u201d GitHub. https:\/\/github.com\/xiaochaihu2002\/cooperation-coevolution-simulations."}],"container-title":["Chaos: An Interdisciplinary Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pubs.aip.org\/aip\/cha\/article-pdf\/doi\/10.1063\/5.0354201\/21159224\/103102_1_5.0354201.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/pubs.aip.org\/aip\/cha\/article-pdf\/doi\/10.1063\/5.0354201\/21159224\/103102_1_5.0354201.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T13:05:52Z","timestamp":1790859952000},"score":1,"resource":{"primary":{"URL":"https:\/\/pubs.aip.org\/cha\/article\/36\/10\/103102\/3405794\/A-novel-route-for-saving-cooperation-on-dynamical"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,10,1]]},"references-count":51,"journal-issue":{"issue":"10","published-print":{"date-parts":[[2026,10,1]]}},"URL":"https:\/\/doi.org\/10.1063\/5.0354201","relation":{},"ISSN":["1054-1500","1089-7682"],"issn-type":[{"value":"1054-1500","type":"print"},{"value":"1089-7682","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2026,10]]},"published":{"date-parts":[[2026,10,1]]},"article-number":"103102"}}