{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,8]],"date-time":"2025-11-08T12:19:35Z","timestamp":1762604375665,"version":"build-2065373602"},"reference-count":105,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,2,28]],"date-time":"2023-02-28T00:00:00Z","timestamp":1677542400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62073202","tsqn201909076"],"award-info":[{"award-number":["62073202","tsqn201909076"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Young Experts of Taishan Scholar Project","award":["62073202","tsqn201909076"],"award-info":[{"award-number":["62073202","tsqn201909076"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Games"],"abstract":"<jats:p>In this paper, a detailed survey is presented for the analysis and control of networked evolutionary games via the matrix method. The algebraic form of networked evolutionary games is firstly recalled. Then, some existing results on networked evolutionary games are summarized. Furthermore, several generalized forms of networked evolutionary games are reviewed, including networked evolutionary games with time delay, networked evolutionary games with bankruptcy mechanism, networked evolutionary games with time-varying networks, and random evolutionary Boolean games. The computational complexity of general networked evolutionary games is still challenging, which limits the application of the matrix method to large-scale networked evolutionary games. Future works are finally presented in the conclusion.<\/jats:p>","DOI":"10.3390\/g14020022","type":"journal-article","created":{"date-parts":[[2023,3,1]],"date-time":"2023-03-01T01:36:09Z","timestamp":1677634569000},"page":"22","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Matrix-Based Method for the Analysis and Control of Networked Evolutionary Games: A Survey"],"prefix":"10.3390","volume":"14","author":[{"given":"Xinrong","family":"Yang","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China"}]},{"given":"Zhenping","family":"Geng","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5322-5375","authenticated-orcid":false,"given":"Haitao","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1038\/246015a0","article-title":"The logic of animal conflict","volume":"246","author":"Smith","year":"1973","journal-title":"Nature"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0022-5193(64)90038-4","article-title":"The genetical evolution of social behaviour. I","volume":"7","author":"Hamilton","year":"1964","journal-title":"J. Theor. Biol."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Fisher, R. (1930). The Genetic Theory of Natural Selection, Clarendon Press.","DOI":"10.5962\/bhl.title.27468"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1086\/406755","article-title":"The evolution of reciprocal altruism","volume":"46","author":"Trivers","year":"1971","journal-title":"Q. Rev. Biol."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2470","DOI":"10.1109\/TWC.2013.031813.121135","article-title":"Joint spectrum sensing and access evolutionary game in cognitive radio networks","volume":"12","author":"Jiang","year":"2013","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1016\/S1874-8651(10)60008-7","article-title":"Dynamic decision model in evolutionary games based on reinforcement learning","volume":"29","author":"Liu","year":"2009","journal-title":"Syst. Eng.-Theory Pract."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"502","DOI":"10.1038\/nature04605","article-title":"A simple rule for the evolution of cooperation on graphs and social networks","volume":"441","author":"Ohtsuki","year":"2006","journal-title":"Nature"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1109\/LCOMM.2012.122012.122676","article-title":"A variable-population evolutionary game model for resource allocation in cooperative cognitive relay networks","volume":"17","author":"Zhang","year":"2013","journal-title":"IEEE Commun. Lett."},{"key":"ref_9","unstructured":"Hofbauer, J., and Sigmund, K. (1988). Evolutionary Games and Population Dynamics, Cambridge University Press."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/0025-5564(78)90077-9","article-title":"Evolutionary stable strategies and game dynamics","volume":"40","author":"Taylor","year":"1978","journal-title":"Math. Biosci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"426","DOI":"10.1016\/j.tpb.2007.12.005","article-title":"Evolutionary stability of first-order information indirect reciprocity in sizable groups","volume":"73","author":"Suzuki","year":"2008","journal-title":"Theor. Popul. Biol."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"995","DOI":"10.1109\/LCOMM.2014.2320894","article-title":"On the equivalence of evolutionary stable strategies","volume":"18","author":"Jiang","year":"2014","journal-title":"IEEE Commun. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1016\/j.cam.2007.06.025","article-title":"An algorithm of evolutionarily stable strategies for the single-population evolutionary game","volume":"217","author":"Lin","year":"2008","journal-title":"J. Comput. Appl. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"126204","DOI":"10.1016\/j.amc.2021.126204","article-title":"Replicator dynamics of an N-player snowdrift game with delayed payoffs","volume":"404","author":"Wettergren","year":"2021","journal-title":"Appl. Math. Comput."},{"key":"ref_15","unstructured":"Weibull, W.J. (1995). Evolutionary Game Theory, MIT Press."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1801","DOI":"10.1109\/TAC.2014.2384755","article-title":"Game interactions and dynamics on networked populations","volume":"60","author":"Madeo","year":"2015","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1016\/j.physrep.2007.04.004","article-title":"Evolutionary games on graphs","volume":"446","author":"Szabo","year":"2007","journal-title":"Phys. Rep."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"826","DOI":"10.1038\/359826a0","article-title":"Evolutionary games and spatial chaos","volume":"359","author":"Nowak","year":"1992","journal-title":"Nature"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"643","DOI":"10.1038\/nature02360","article-title":"Spatial structure often inhibits the evolution of cooperation in the snowdrift game","volume":"428","author":"Hauert","year":"2004","journal-title":"Nature"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"063013","DOI":"10.1088\/1367-2630\/ab241b","article-title":"Evolutionary dynamics of networked multi-person games: Mixing opponent-aware and opponent-independent strategy decisions","volume":"21","author":"Huang","year":"2019","journal-title":"New J. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1038\/nature06940","article-title":"Social diversity promotes the emergence of cooperation in public goods games","volume":"454","author":"Santos","year":"2008","journal-title":"Nature"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1103\/PhysRevE.58.69","article-title":"Evolutionary prisoner\u2019s dilemma game on a square lattice","volume":"58","author":"Szabo","year":"1998","journal-title":"Phys. Rev. E"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"056113","DOI":"10.1103\/PhysRevE.74.056113","article-title":"Memory-based snowdrift game on networks","volume":"74","author":"Wang","year":"2006","journal-title":"Phys. Rev. E"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"4502","DOI":"10.1016\/j.physa.2011.07.021","article-title":"The influence of tie strength on evolutionary games on networks: An empirical investigation","volume":"390","author":"Buesser","year":"2011","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_25","first-page":"1","article-title":"Evolutionary games on complex networks","volume":"2","author":"Wang","year":"2007","journal-title":"CAAI Trans. Intell. Syst."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Cheng, D., Qi, H., and Li, Z. (2011). Analysis and Control of Boolean Networks: A Semi-Tensor Product Approach, Springer.","DOI":"10.1007\/978-0-85729-097-7"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"010204","DOI":"10.1007\/s11432-017-9269-6","article-title":"Nonsingularity of Grain-like cascade FSRs via semi-tensor product","volume":"61","author":"Lu","year":"2018","journal-title":"Sci. China Inf. Sci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1506","DOI":"10.1016\/j.automatica.2013.02.027","article-title":"On the periodic trajectories of Boolean control networks","volume":"49","author":"Fornasini","year":"2013","journal-title":"Automatica"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"6093","DOI":"10.1109\/TAC.2021.3060733","article-title":"Asymptotic stability of Boolean networks with multiple missing data","volume":"66","author":"Huang","year":"2021","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1016\/j.automatica.2015.08.006","article-title":"Set stability and set stabilization of Boolean control networks based on invariant subsets","volume":"61","author":"Guo","year":"2015","journal-title":"Automatica"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"3701","DOI":"10.1016\/j.nonrwa.2011.07.003","article-title":"Stability and stabilization of multivalued logical networks","volume":"12","author":"Li","year":"2011","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1585","DOI":"10.1109\/TNNLS.2015.2449274","article-title":"Pinning control sesign for the stabilization of Boolean networks","volume":"27","author":"Li","year":"2016","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1016\/j.automatica.2016.12.007","article-title":"Set stabilization for switched Boolean control networks","volume":"78","author":"Li","year":"2017","journal-title":"Automatica"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1168","DOI":"10.1016\/j.sysconle.2013.09.008","article-title":"Simultaneous stabilization for a set of Boolean control networks","volume":"62","author":"Li","year":"2013","journal-title":"Syst. Control Lett."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"4385","DOI":"10.1137\/18M1169308","article-title":"Stabilization of Boolean control networks under aperiodic sampled-data control","volume":"56","author":"Lu","year":"2018","journal-title":"SIAM J. Control Optim."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1658","DOI":"10.1109\/TAC.2015.2478123","article-title":"On pinning controllability of Boolean control networks","volume":"61","author":"Lu","year":"2016","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.amc.2019.03.059","article-title":"On the ensemble controllability of Boolean control networks using STP method","volume":"358","author":"Zhong","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1109\/TNNLS.2020.3027599","article-title":"Finite-horizon optimal control of Boolean control networks: A unified graph-theoretical approach","volume":"33","author":"Gao","year":"2022","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1766","DOI":"10.1109\/TAC.2010.2092290","article-title":"Optimal control of logical control networks","volume":"56","author":"Zhao","year":"2011","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2314","DOI":"10.1109\/TCYB.2020.3003552","article-title":"Infinite-horizon optimal control of switched Boolean control networks with average cost: An efficient graph-theoretical approach","volume":"52","author":"Gao","year":"2022","journal-title":"IEEE Trans. Cybern."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"092201","DOI":"10.1007\/s11432-017-9135-4","article-title":"Observability of Boolean control networks","volume":"61","author":"Zhu","year":"2018","journal-title":"Sci. China Inf. Sci."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"188","DOI":"10.1016\/j.cam.2017.06.016","article-title":"Control design for output tracking of delayed Boolean control networks","volume":"327","author":"Liu","year":"2018","journal-title":"J. Comput. Appl. Math."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Mu, T., Feng, J., and Li, Y. (2021, January 26\u201328). Controllability and reachability of k-valued logical control networks with time delays in states. Proceedings of the 40th Chinese Control Conference, Shanghai, China.","DOI":"10.23919\/CCC52363.2021.9550150"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"7812","DOI":"10.1016\/j.jfranklin.2017.09.024","article-title":"Stabilization and set. stabilization of delayed Boolean control networks based on trajectory stabilization","volume":"354","author":"Zheng","year":"2017","journal-title":"J. Frankl. Inst."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1272","DOI":"10.1016\/j.automatica.2014.02.034","article-title":"State feedback stabilization for probabilistic Boolean networks","volume":"50","author":"Li","year":"2014","journal-title":"Automatica"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"340","DOI":"10.1016\/j.automatica.2014.12.018","article-title":"Controllability of probabilistic Boolean control networks based on transition probability matrices","volume":"52","author":"Liu","year":"2015","journal-title":"Automatica"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"2917","DOI":"10.1016\/j.automatica.2012.08.029","article-title":"On reachability and controllability of switched Boolean control networks","volume":"48","author":"Li","year":"2012","journal-title":"Automatica"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"3632","DOI":"10.1137\/19M1281332","article-title":"Perturbation analysis for controllability of logical control networks","volume":"58","author":"Li","year":"2020","journal-title":"SIAM J. Control Optim."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"4623","DOI":"10.1109\/TCYB.2020.3003055","article-title":"Perturbation analysis for finite-time stability and stabilization of probabilistic Boolean networks","volume":"51","author":"Li","year":"2021","journal-title":"IEEE Trans. Cybern."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"101216","DOI":"10.1016\/j.nahs.2022.101216","article-title":"Robust stability of switched Boolean networks with function perturbation","volume":"46","author":"Wu","year":"2022","journal-title":"Nonlinear Anal. Hybrid Syst."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"1198","DOI":"10.1109\/TSMC.2022.3195196","article-title":"Reachability, controllability and stabilization of Boolean control networks with stochastic function perturbations","volume":"53","author":"Yang","year":"2022","journal-title":"IEEE Trans. Syst. Man, Cybern. Syst."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"2402","DOI":"10.1109\/TAC.2015.2404471","article-title":"Modeling, Analysis and Control of Networked Evolutionary Games","volume":"60","author":"Cheng","year":"2015","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"010201","DOI":"10.1007\/s11432-017-9265-2","article-title":"From STP to game-based control","volume":"61","author":"Cheng","year":"2018","journal-title":"Sci. China Inf. Sci."},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Ge, M., Zhao, J., Xing, H., and Wang, J. (2016, January 27\u201329). Impact of social punishment on networked evolutionary games via semi-tensor product method. Proceedings of the 35th Chinese Control Conference, Chengdu, China.","DOI":"10.1109\/ChiCC.2016.7553079"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1007\/s11633-017-1056-0","article-title":"Dynamics and stability of potential hyper-networked evolutionary games","volume":"14","author":"Liu","year":"2017","journal-title":"Int. J. Autom. Comput."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1109\/JAS.2014.7004614","article-title":"On dynamics and Nash equilibriums of networked games","volume":"1","author":"Cheng","year":"2014","journal-title":"IEEE\/CAA J. Autom. Sin."},{"key":"ref_57","first-page":"11","article-title":"Topological structure of graph-based networked evolutionary games","volume":"56","author":"Cheng","year":"2021","journal-title":"J. Shandong Univ. Nat. Sci."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"1793","DOI":"10.1016\/j.automatica.2014.05.005","article-title":"On finite potential game","volume":"50","author":"Cheng","year":"2014","journal-title":"Automatica"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"052204","DOI":"10.1007\/s11432-015-5376-9","article-title":"Stable degree analysis for strategy profifiles of evolutionary networked games","volume":"59","author":"Guo","year":"2016","journal-title":"Sci. China Inf. Sci."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"2548","DOI":"10.1080\/00207721.2018.1506063","article-title":"Modelling and strategy consensus for a class of networked evolutionary games","volume":"49","author":"Zhao","year":"2018","journal-title":"Int. J. Syst. Sci."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"126710","DOI":"10.1016\/j.physa.2021.126710","article-title":"Modeling and optimization for networked evolutionary games with player exit mechanism: Semi-tensor product of matrices method","volume":"590","author":"Tang","year":"2022","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_62","first-page":"279","article-title":"Dynamics and optimization of control networked evolutionary games with local information","volume":"36","author":"Wang","year":"2019","journal-title":"Control Theory Appl."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"1335","DOI":"10.1109\/TNNLS.2013.2293149","article-title":"Evolutionarily stable strategy of networked evolutionary games","volume":"25","author":"Cheng","year":"2014","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1016\/j.jtbi.2006.06.004","article-title":"The replicator equation on graphs","volume":"243","author":"Ohtsuki","year":"2006","journal-title":"J. Theor. Biol."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"442","DOI":"10.1016\/j.amc.2018.05.027","article-title":"Stabilization of evolutionary networked games with length-r information","volume":"337","author":"Mao","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"ref_66","doi-asserted-by":"crossref","unstructured":"Wang, Y., and Cheng, D. (2015, January 23\u201325). Dynamics and stability of evolutionary games with time-invariant delay in strategies. Proceedings of the 27th Chinese Control and Decision Conference, Qingdao, China.","DOI":"10.1109\/CCDC.2015.7161975"},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"818","DOI":"10.1109\/JAS.2016.7510259","article-title":"A matrix approach to the modeling and analysis of networked evolutionary games with time delays","volume":"5","author":"Zhao","year":"2018","journal-title":"IEEE\/CAA J. Autom. Sin."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"1778","DOI":"10.1109\/TCNS.2021.3084548","article-title":"Modeling and dynamics of networked evolutionary game with switched time delay","volume":"8","author":"Zheng","year":"2021","journal-title":"IEEE Trans. Control Netw. Syst."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"1104","DOI":"10.1080\/00207179.2017.1306113","article-title":"Modelling and strategy optimisation for a kind of networked evolutionary games with memories under the bankruptcy mechanism","volume":"91","author":"Fu","year":"2017","journal-title":"Int. J. Control"},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"717","DOI":"10.1002\/asjc.1412","article-title":"A matrix approach to the analysis and control of networked evolutionary games with bankruptcy mechanism","volume":"19","author":"Fu","year":"2017","journal-title":"Asian J. Control"},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"125806","DOI":"10.1109\/ACCESS.2020.3008198","article-title":"Optimal control of networked evolutionary games with bankruptcy risk","volume":"8","author":"Li","year":"2020","journal-title":"IEEE Access"},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"033112","DOI":"10.1063\/1.3621719","article-title":"Cascading failures and the emergence of cooperation in evolutionary-game based models of social and economical networks","volume":"21","author":"Wang","year":"2011","journal-title":"Chaos"},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"3093","DOI":"10.1007\/s00034-017-0707-2","article-title":"Model and control for a class of networked evolutionary games with finite memories and time-varying networks","volume":"37","author":"Fu","year":"2018","journal-title":"Circuits, Syst. Signal Process."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"109378","DOI":"10.1016\/j.knosys.2022.109378","article-title":"Modeling and optimization control of networked evolutionary games with heterogeneous memories and switched topologies","volume":"252","author":"Yuan","year":"2022","journal-title":"Knowl.-Based Syst."},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"98","DOI":"10.1016\/j.nahs.2016.03.009","article-title":"Formulation and optimization control of a class of networked evolutionary games with switched topologies","volume":"22","author":"Zhao","year":"2016","journal-title":"Nonlinear Anal. Hybrid Syst."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"107644","DOI":"10.1016\/j.knosys.2021.107644","article-title":"Strategy optimization of weighted networked evolutionary games with switched topologies and threshold","volume":"235","author":"Zhu","year":"2022","journal-title":"Knowl.-Based Syst."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.amc.2017.02.020","article-title":"Stochastic stability and stabilization of n-person random evolutionary Boolean games","volume":"306","author":"Ding","year":"2017","journal-title":"Appl. Math. Comput."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"144","DOI":"10.1080\/00207179.2019.1585957","article-title":"Optimal control of random evolutionary Boolean games","volume":"94","author":"Ding","year":"2019","journal-title":"Int. J. Control"},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"2152","DOI":"10.1049\/iet-cta.2017.0047","article-title":"Stochastic set stabilisation of n-person random evolutionary boolean games and its applications","volume":"11","author":"Li","year":"2017","journal-title":"IET Control Theory Appl."},{"key":"ref_80","doi-asserted-by":"crossref","unstructured":"Tang, Y., Li, L., and Lu, J. (2020, January 6\u20138). Modeling and optimization of a class of networked evolutionary games with random entrance and time delays. Proceedings of the 2020 Chinese Automation Congress, Shanghai, China.","DOI":"10.1109\/CAC51589.2020.9327005"},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1016\/j.amc.2016.05.012","article-title":"A matrix approach to modeling and optimization for dynamic games with random entrance","volume":"290","author":"Zhao","year":"2016","journal-title":"Appl. Math. Comput."},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"57","DOI":"10.2307\/2951778","article-title":"The evolution of conventions","volume":"61","author":"Young","year":"1993","journal-title":"Econometrica"},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"011909","DOI":"10.1103\/PhysRevE.74.011909","article-title":"Stochastic dynamics of invasion and fixation","volume":"74","author":"Traulsen","year":"2006","journal-title":"Phys. Rev. E"},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"889","DOI":"10.1016\/j.jet.2016.03.010","article-title":"Graphical potential games","volume":"163","author":"Babichenko","year":"2016","journal-title":"J. Econ. Theory"},{"key":"ref_85","unstructured":"Fudenberg, D., and Levine, D.K. (1998). The Theory of Learning in Games, MIT Press."},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/j.cam.2018.10.003","article-title":"Algebraic formulation and topological structure of Boolean networks with state-dependent delay","volume":"350","author":"Li","year":"2019","journal-title":"J. Comput. Appl. Math."},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"7735","DOI":"10.1016\/j.jfranklin.2018.08.009","article-title":"Set stability and synchronization of logical networks with probabilistic time delays","volume":"355","author":"Ding","year":"2018","journal-title":"J. Frankl. Inst."},{"key":"ref_88","first-page":"790","article-title":"Stability and stabilization of Boolean networks with stochastic delays","volume":"64","author":"Meng","year":"2019","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"1819","DOI":"10.1007\/s12555-020-0204-4","article-title":"Time-variant feedback stabilization of constrained delayed Boolean networks under nonuniform sampled-data control","volume":"19","author":"Kong","year":"2021","journal-title":"Int. J. Control. Autom. Syst."},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"169203","DOI":"10.1007\/s11432-019-9904-6","article-title":"State-feedback set stabilization of logical control networks with state-dependent delay","volume":"64","author":"Zheng","year":"2021","journal-title":"Sci. China Inf. Sci."},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1016\/j.neucom.2013.09.012","article-title":"Controllability of Boolean control networks with time delays both in states and inputs","volume":"129","author":"Han","year":"2014","journal-title":"Neurocomputing"},{"key":"ref_92","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1016\/j.automatica.2011.01.040","article-title":"Controllability of Boolean control networks with time delays in states","volume":"47","author":"Li","year":"2011","journal-title":"Automatica"},{"key":"ref_93","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1137\/140991820","article-title":"On controllability of delayed Boolean control networks","volume":"54","author":"Lu","year":"2016","journal-title":"SIAM J. Control Optim."},{"key":"ref_94","doi-asserted-by":"crossref","first-page":"108201","DOI":"10.1007\/s11432-012-4651-2","article-title":"Controllability of time-variant Boolean control networks and its application to Boolean control networks with finite memories","volume":"56","author":"Zhang","year":"2013","journal-title":"Sci. China Inf. Sci."},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"1478","DOI":"10.1109\/TNNLS.2013.2246187","article-title":"Controllability and observability of Boolean control networks with time-variant delays in states","volume":"24","author":"Zhang","year":"2013","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_96","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1007\/s11424-017-6145-1","article-title":"Observability of Boolean control networks with time-variant delays in states","volume":"31","author":"Jiang","year":"2018","journal-title":"J. Syst. Sci. Complex."},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"948","DOI":"10.1109\/TNN.2011.2126594","article-title":"Observability of Boolean control networks with state time delays","volume":"22","author":"Li","year":"2011","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_98","doi-asserted-by":"crossref","first-page":"50007","DOI":"10.1209\/0295-5075\/84\/50007","article-title":"Making new connections towards cooperation in the prisoner\u2019s dilemma game","volume":"84","author":"Szolnoki","year":"2008","journal-title":"Europhys. Lett."},{"key":"ref_99","doi-asserted-by":"crossref","first-page":"e01169","DOI":"10.7554\/eLife.01169","article-title":"Spatial dilemmas of diffusible public goods","volume":"2","author":"Allen","year":"2013","journal-title":"eLife"},{"key":"ref_100","doi-asserted-by":"crossref","first-page":"845","DOI":"10.1038\/srep00845","article-title":"The evolution of homophily","volume":"2","author":"Fu","year":"2012","journal-title":"Sci. Rep."},{"key":"ref_101","doi-asserted-by":"crossref","first-page":"637","DOI":"10.1007\/s10955-012-0631-6","article-title":"Global migration can lead to stronger spatial selection than local migration","volume":"151","author":"Fu","year":"2013","journal-title":"J. Stat. Phys."},{"key":"ref_102","doi-asserted-by":"crossref","first-page":"5144","DOI":"10.1109\/TAC.2020.2968836","article-title":"Efficient verification of observability and reconstructibility for large Boolean control networks with special structures","volume":"65","author":"Zhang","year":"2020","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_103","doi-asserted-by":"crossref","first-page":"1105","DOI":"10.1016\/j.physleta.2009.02.004","article-title":"Evolution of a large online social network","volume":"373","author":"Hu","year":"2009","journal-title":"Phys. Lett. A"},{"key":"ref_104","doi-asserted-by":"crossref","first-page":"841","DOI":"10.1209\/epl\/i2006-10217-3","article-title":"Chaos promotes cooperation in the spatial prisoner\u2019s dilemma game","volume":"75","author":"Perc","year":"2006","journal-title":"Europhys. Lett."},{"key":"ref_105","doi-asserted-by":"crossref","first-page":"022101","DOI":"10.1103\/PhysRevE.75.022101","article-title":"Transition from Gaussian to L\u00e9vy distributions of stochastic payoff variations in the spatial prisoner\u2019s dilemma game","volume":"75","author":"Perc","year":"2007","journal-title":"Phys. Rev. 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