{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:36:04Z","timestamp":1760060164959,"version":"build-2065373602"},"reference-count":47,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,8,9]],"date-time":"2025-08-09T00:00:00Z","timestamp":1754697600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"STI 2030\u2014Major Projects","award":["2022ZD0211400","Nos. T2293771","62306191"],"award-info":[{"award-number":["2022ZD0211400","Nos. T2293771","62306191"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2022ZD0211400","Nos. T2293771","62306191"],"award-info":[{"award-number":["2022ZD0211400","Nos. T2293771","62306191"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"New Cornerstone Science Foundation through the XPLORER PRIZE","award":["2022ZD0211400","Nos. T2293771","62306191"],"award-info":[{"award-number":["2022ZD0211400","Nos. T2293771","62306191"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Research on network robustness has long focused on changes in the structure connectivity of networks under attacks, effectively depicting structural integrity while ignoring the exploration of functional integrity. When the core path of the network is attacked, even if it remains connected, the rapid increase in energy consumption may still trigger systematic risks. Existing studies mainly use random networks and scale-free networks as comparative models, which has become a classic research paradigm. However, real-world networks often exhibit mixed topological features. To address the above issues, this paper introduces the concept of energy from physics into bipartite networks and establishes an evaluation framework for assessing the synergistic effects of structural evolution and attack strategies on network matching robustness. We first introduce a structural parameter u to construct a structural evolution model, where the network\u2019s minimal matching energy distribution evolves from topological heterogeneity to random features. When u approaches 0, edges with the minimal matching energy concentrate on a few candidates, manifesting scale-free network features. When u approaches 1, the uniform distribution of the minimum-matching-energy edges corresponds to random network features. We then design three types of edge attack strategies\u2014minimum-energy (min-E), random-energy (ran-E), and maximum-energy (max-E) attacks\u2014simulating the impacts of critical path destruction, uniform perturbation, and redundancy removal, respectively. In addition, we construct two evaluation indicators, the average matching energy and the matching retention rate. The results show that structural evolution significantly affects network matching robustness in a nonlinear manner. Different attack strategies also exert different influence on matching robustness. Furthermore, the findings reveal the synergistic effects of the two factors on network matching robustness. The synergistic effects of redundancy capacity and network structure on matching robustness are also explored. The research deepens the understanding of network matching robustness and provides a theoretical basis for resource allocation systems to combat network attacks.<\/jats:p>","DOI":"10.3390\/e27080847","type":"journal-article","created":{"date-parts":[[2025,8,11]],"date-time":"2025-08-11T11:02:09Z","timestamp":1754910129000},"page":"847","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Synergistic Effects of Structural Evolution and Attack Strategies on Network Matching Robustness"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4038-1410","authenticated-orcid":false,"given":"Xu","family":"Na","sequence":"first","affiliation":[{"name":"Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China"},{"name":"Department of Physics, University of Fribourg, 1700 Fribourg, Switzerland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0006-0307-5800","authenticated-orcid":false,"given":"Junying","family":"Cui","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Fribourg, 1700 Fribourg, Switzerland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0006-1895-9000","authenticated-orcid":false,"given":"Chang","family":"Su","sequence":"additional","affiliation":[{"name":"Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2089-5150","authenticated-orcid":false,"given":"Shimin","family":"Cai","sequence":"additional","affiliation":[{"name":"School of Computer Science and Engineering (School of Cyber Security), University of Electronic Science and Technology of China, Chengdu 611731, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2156-0432","authenticated-orcid":false,"given":"Linyuan","family":"L\u00fc","sequence":"additional","affiliation":[{"name":"School of Cyber Science and Technology, University of Science and Technology of China, Hefei 230026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"4176","DOI":"10.1109\/TSMC.2023.3241621","article-title":"Enhancing the robustness of networks against multiple damage models using a multifactorial evolutionary algorithm","volume":"53","author":"Wang","year":"2023","journal-title":"IEEE Trans. Syst. Man, Cybern. Syst."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Zhao, J., Li, D., Sanhedrai, H., Cohen, R., and Havlin, S. (2016). Spatio-temporal propagation of cascading overload failures in spatially embedded networks. Nat. Commun., 7.","DOI":"10.1038\/ncomms10094"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Dong, G., Gao, J., Du, R., Tian, L., Stanley, H.E., and Havlin, S. (2013). Robustness of network of networks under targeted attack. Phys. Rev. E Stat. Nonlinear Soft Matter Phys., 87.","DOI":"10.1103\/PhysRevE.87.052804"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Li, Z., Liu, J., Ren, J., Dong, Y., and Li, W. (2024). Application of hybrid strategies of complex network attack and defense games. Chaos Solitons Fractals, 189.","DOI":"10.1016\/j.chaos.2024.115662"},{"key":"ref_5","first-page":"5915","article-title":"Graph vulnerability and robustness: A survey","volume":"35","author":"Freitas","year":"2022","journal-title":"IEEE Trans. Knowl. Data Eng."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"568","DOI":"10.1007\/s11704-016-6108-z","article-title":"A comparative study of network robustness measures","volume":"11","author":"Liu","year":"2017","journal-title":"Front. Comput. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"6244","DOI":"10.1109\/TSMC.2019.2960156","article-title":"Robustness evaluation of multipartite complex networks based on percolation theory","volume":"51","author":"Cai","year":"2020","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1038\/s42254-023-00676-y","article-title":"Robustness and resilience of complex networks","volume":"6","author":"Artime","year":"2024","journal-title":"Nat. Rev. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"175702","DOI":"10.1103\/PhysRevLett.101.175702","article-title":"Wave localization in complex networks with high clustering","volume":"101","author":"Jahnke","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"888","DOI":"10.1038\/nphys1746","article-title":"Identification of influential spreaders in complex networks","volume":"6","author":"Kitsak","year":"2010","journal-title":"Nat. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"821","DOI":"10.1109\/TSMC.2017.2733545","article-title":"Subgraph robustness of complex networks under attacks","volume":"49","author":"Shang","year":"2017","journal-title":"IEEE Trans. Syst. Man, Cybern. Syst."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/j.physa.2016.03.075","article-title":"The dynamic correlation between degree and betweenness of complex network under attack","volume":"457","author":"Nie","year":"2016","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1025","DOI":"10.1038\/nature08932","article-title":"Catastrophic cascade of failures in interdependent networks","volume":"464","author":"Buldyrev","year":"2010","journal-title":"Nature"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Gu, Y., Fu, X., Liu, Z., Xu, X., and Chen, A. (2020). Performance of transportation network under perturbations: Reliability, vulnerability, and resilience. Transp. Res. Part E Logist. Transp. Rev., 133.","DOI":"10.1016\/j.tre.2019.11.003"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1007\/s12469-018-00193-7","article-title":"Vulnerability of public transportation networks against directed attacks and cascading failures","volume":"11","author":"Candelieri","year":"2019","journal-title":"Public Transp."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1395","DOI":"10.1007\/s10618-015-0447-5","article-title":"Optimizing network robustness by edge rewiring: A general framework","volume":"30","author":"Chan","year":"2016","journal-title":"Data Min. Knowl. Discov."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"12730","DOI":"10.1016\/j.eswa.2011.04.061","article-title":"A distribution network optimization problem for third party logistics service providers","volume":"38","author":"Kara","year":"2011","journal-title":"Expert Syst. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ress.2018.02.008","article-title":"Simulation-based catastrophe-induced port loss estimation","volume":"175","author":"Cao","year":"2018","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1007\/s11067-019-09482-5","article-title":"Disruptions in spatial networks: A comparative study of major shocks affecting ports and shipping patterns","volume":"20","author":"Rousset","year":"2020","journal-title":"Netw. Spat. Econ."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.retrec.2009.12.004","article-title":"Foreland-based regionalization: Integrating intermediate hubs with port hinterlands","volume":"27","author":"Rodrigue","year":"2010","journal-title":"Res. Transp. Econ."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Erhardt, G.D., Roy, S., Cooper, D., Sana, B., Chen, M., and Castiglione, J. (2019). Do transportation network companies decrease or increase congestion?. Sci. Adv., 5.","DOI":"10.1126\/sciadv.aau2670"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Sun, X., and Wandelt, S. (2021). Robustness of air transportation as complex networks: Systematic review of 15 years of research and outlook into the future. Sustainability, 13.","DOI":"10.3390\/su13116446"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"494","DOI":"10.1038\/s41893-020-00678-z","article-title":"Impacts of transportation network companies on urban mobility","volume":"4","author":"Diao","year":"2021","journal-title":"Nat. Sustain."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Ficara, A., Curreri, F., Fiumara, G., De Meo, P., and Liotta, A. (2022). Covert network construction, disruption, and resilience: A survey. Mathematics, 10.","DOI":"10.3390\/math10162929"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Duijn, P.A., Kashirin, V., and Sloot, P.M. (2014). The relative ineffectiveness of criminal network disruption. Sci. Rep., 4.","DOI":"10.1038\/srep04238"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Baybusinov, I.B., Fenoaltea, E.M., and Zhang, Y.C. (2022). Negotiation problem. Phys. A Stat. Mech. Appl., 592.","DOI":"10.1016\/j.physa.2021.126806"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/S0378-4371(02)01953-2","article-title":"Matching games with partial information","volume":"324","author":"Laureti","year":"2003","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2021.03.001","article-title":"The stable marriage problem: An interdisciplinary review from the physicist\u2019s perspective","volume":"917","author":"Fenoaltea","year":"2021","journal-title":"Phys. Rep."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Shang, Y. (2021). Generalized k-cores of networks under attack with limited knowledge. Chaos Solitons Fractals, 152.","DOI":"10.1016\/j.chaos.2021.111305"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Yin, Y., Liu, Q., Zhang, C., and Zhou, J. (2019). Survivability analysis of weighted-edge attacks on complex networks with incomplete information. Phys. A Stat. Mech. Appl., 531.","DOI":"10.1016\/j.physa.2019.04.193"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"665","DOI":"10.1126\/science.1216142","article-title":"Critical truths about power laws","volume":"335","author":"Stumpf","year":"2012","journal-title":"Science"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"17268","DOI":"10.1073\/pnas.0806977105","article-title":"Universality of citation distributions: Toward an objective measure of scientific impact","volume":"105","author":"Radicchi","year":"2008","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"14812","DOI":"10.1073\/pnas.1918901117","article-title":"Power-law distribution of degree\u2013degree distance: A better representation of the scale-free property of complex networks","volume":"117","author":"Zhou","year":"2020","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Broido, A.D., and Clauset, A. (2019). Scale-free networks are rare. Nat. Commun., 10.","DOI":"10.1038\/s41467-019-08746-5"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1126\/science.286.5439.509","article-title":"Emergence of scaling in random networks","volume":"286","author":"Albert","year":"1999","journal-title":"Science"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Chakrabarti, D., and Faloutsos, C. (2006). Graph mining: Laws, generators, and algorithms. ACM Comput. Surv. (CSUR), 38.","DOI":"10.1145\/1132952.1132954"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Omar, Y.M., and Plapper, P. (2020). A survey of information entropy metrics for complex networks. Entropy, 22.","DOI":"10.3390\/e22121417"},{"key":"ref_39","unstructured":"Cheng, W., H\u00fcllermeier, E., and Dembczynski, K.J. (2010, January 21\u201325). Label ranking methods based on the Plackett-Luce model. Proceedings of the 27th International Conference on Machine Learning (ICML-10), Haifa, Israel."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Guiver, J., and Snelson, E. (2009, January 14\u201318). Bayesian inference for Plackett-Luce ranking models. Proceedings of the 26th Annual International Conference on Machine Learning, Montreal, QC, Canada.","DOI":"10.1145\/1553374.1553423"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Hadji, M., and Zeghlache, D. (2012, January 24\u201329). Minimum cost maximum flow algorithm for dynamic resource allocation in clouds. Proceedings of the 2012 IEEE Fifth International Conference on Cloud Computing, Honolulu, HI, USA.","DOI":"10.1109\/CLOUD.2012.36"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1145\/3610940","article-title":"Almost-linear-time algorithms for maximum flow and minimum-cost flow","volume":"66","author":"Chen","year":"2023","journal-title":"Commun. ACM"},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Cicchini, T., Caridi, I., and Ermann, L. (2024). Robustness of the public transport network against attacks on its routes. Chaos Solitons Fractals, 184.","DOI":"10.1016\/j.chaos.2024.115019"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"4626","DOI":"10.1109\/JSYST.2020.2964624","article-title":"A stochastic model of cascading failure dynamics in cyber-physical power systems","volume":"14","author":"Gao","year":"2020","journal-title":"IEEE Syst. J."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Mizutaka, S., and Yakubo, K. (2015). Robustness of scale-free networks to cascading failures induced by fluctuating loads. Phys. Rev. E, 92.","DOI":"10.1103\/PhysRevE.92.012814"},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Radicchi, F., and Bianconi, G. (2017). Redundant interdependencies boost the robustness of multiplex networks. Phys. Rev. X, 7.","DOI":"10.1103\/PhysRevX.7.011013"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"6385","DOI":"10.1080\/00207543.2019.1566666","article-title":"A graph-based model to measure structural redundancy for supply chain resilience","volume":"57","author":"Tan","year":"2019","journal-title":"Int. J. Prod. Res."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/8\/847\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:27:14Z","timestamp":1760034434000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/8\/847"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,9]]},"references-count":47,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2025,8]]}},"alternative-id":["e27080847"],"URL":"https:\/\/doi.org\/10.3390\/e27080847","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2025,8,9]]}}}