{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T07:24:21Z","timestamp":1760081061508,"version":"build-2065373602"},"reference-count":48,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,3,28]],"date-time":"2025-03-28T00:00:00Z","timestamp":1743120000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Key R&amp;D Program of China","award":["2022YFC3202602","B230201057","U2340221","52079044"],"award-info":[{"award-number":["2022YFC3202602","B230201057","U2340221","52079044"]}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["2022YFC3202602","B230201057","U2340221","52079044"],"award-info":[{"award-number":["2022YFC3202602","B230201057","U2340221","52079044"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2022YFC3202602","B230201057","U2340221","52079044"],"award-info":[{"award-number":["2022YFC3202602","B230201057","U2340221","52079044"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2022YFC3202602","B230201057","U2340221","52079044"],"award-info":[{"award-number":["2022YFC3202602","B230201057","U2340221","52079044"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Identifying the key components in complex systems is crucial for ensuring the functionality and integrity of the system. For directed networks, both nodes and edges are critical components, but little attention has been drawn to the important edges. In this paper, a novel method, Cooperation\u2013Competition Centrality (Co-Com Centrality), has been proposed to quantify the edge importance in a directed network. This method utilizes cooperation and competition relationships to integrate the local and global information about the network. We compare the performance of Co-Com Centrality with five typical methods in nine networks; the experiments show that Co-Com Centrality can accurately identify edges which are crucial for maintaining global network efficiency and network stability, while also accurately assessing the edge importance.<\/jats:p>","DOI":"10.3390\/axioms14040258","type":"journal-article","created":{"date-parts":[[2025,3,28]],"date-time":"2025-03-28T13:36:49Z","timestamp":1743169009000},"page":"258","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Novel Method to Identify Important Edges in a Directed Network"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1435-2907","authenticated-orcid":false,"given":"Saiyu","family":"Yuan","sequence":"first","affiliation":[{"name":"The National Key Laboratory of Water Disaster Prevention, Hohai University, Nanjing 210098, China"},{"name":"Key Laboratory of Hydrologic-Cycle and Hydrodynamic-System of Ministry of Water Resources, Hohai University, Nanjing 210098, China"}]},{"given":"Wenqiang","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Mathematics, Hohai University, Nanjing 210098, China"}]},{"given":"Guanghui","family":"Yan","sequence":"additional","affiliation":[{"name":"The National Key Laboratory of Water Disaster Prevention, Hohai University, Nanjing 210098, China"}]},{"given":"Hongwu","family":"Tang","sequence":"additional","affiliation":[{"name":"The National Key Laboratory of Water Disaster Prevention, Hohai University, Nanjing 210098, China"},{"name":"Key Laboratory of Hydrologic-Cycle and Hydrodynamic-System of Ministry of Water Resources, Hohai University, Nanjing 210098, China"}]},{"given":"Xing","family":"Cheng","sequence":"additional","affiliation":[{"name":"School of Mathematics, Hohai University, Nanjing 210098, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1016\/j.eng.2022.01.015","article-title":"Theory and practice of hydrodynamic reconstruction in plain river networks","volume":"24","author":"Tang","year":"2023","journal-title":"Engineering"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"416","DOI":"10.1016\/j.ecolind.2017.08.049","article-title":"A combined model for river health evaluation based upon the physical, chemical, and biological elements","volume":"84","author":"Zhang","year":"2018","journal-title":"Ecol. Indic."},{"key":"ref_3","first-page":"79","article-title":"Two-Stage Optimization of Urban Water System Connectivity Scheme under Structure-Function Coupling","volume":"33","author":"Dou","year":"2022","journal-title":"Adv. Water Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"117015","DOI":"10.1016\/j.watres.2021.117015","article-title":"New framework for natural-artificial transport paths and hydrological connectivity analysis in an agriculture-intensive catchment","volume":"196","author":"Sun","year":"2021","journal-title":"Water Res."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"119055","DOI":"10.1016\/j.jenvman.2023.119055","article-title":"Assessing hydrological connectivity for natural-artificial catchment with a new framework integrating graph theory and network analysis","volume":"346","author":"Wang","year":"2023","journal-title":"J. Environ. Manag."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1080\/0022250X.1972.9989806","article-title":"Factoring and weighting approaches to status scores and clique identification","volume":"2","author":"Bonacich","year":"1972","journal-title":"J. Math. Sociol."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"10168","DOI":"10.1038\/ncomms10168","article-title":"The H-index of a network node and its relation to degree and coreness","volume":"7","author":"Zhou","year":"2016","journal-title":"Nat. Commun."},{"key":"ref_8","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_9","doi-asserted-by":"crossref","first-page":"1777","DOI":"10.1016\/j.physa.2011.09.017","article-title":"Identifying influential nodes in complex networks","volume":"391","author":"Chen","year":"2012","journal-title":"Phys. Stat. Mech. Its Appl."},{"key":"ref_10","first-page":"1","article-title":"Greedily improving our own closeness centrality in a network","volume":"11","author":"Crescenzi","year":"2016","journal-title":"ACM Trans. Knowl. Discov. Data (TKDD)"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Freeman, L.C. (1977). A set of measures of centrality based on betweenness. Sociometry, 35\u201341.","DOI":"10.2307\/3033543"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/0378-8733(94)00248-9","article-title":"Eccentricity and centrality in networks","volume":"17","author":"Hage","year":"1995","journal-title":"Soc. Networks"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1007\/BF02289026","article-title":"A new status index derived from sociometric analysis","volume":"18","author":"Katz","year":"1953","journal-title":"Psychometrika"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/S0169-7552(98)00110-X","article-title":"The anatomy of a large-scale hypertextual web search engine","volume":"30","author":"Brin","year":"1998","journal-title":"Comput. Netw. ISDN Syst."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Brandes, U., and Fleischer, D. (2005, January 24\u201326). Centrality measures based on current flow. Proceedings of the Annual Symposium on Theoretical Aspects of Computer Science, Stuttgart, Germany.","DOI":"10.1007\/978-3-540-31856-9_44"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1038\/s42005-021-00633-0","article-title":"Unraveling the effects of multiscale network entanglement on empirical systems","volume":"4","author":"Ghavasieh","year":"2021","journal-title":"Commun. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"032812","DOI":"10.1103\/PhysRevE.90.032812","article-title":"Role of centrality for the identification of influential spreaders in complex networks","volume":"90","author":"Barbieri","year":"2014","journal-title":"Phys. Rev. E"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1038\/s41467-023-44257-0","article-title":"DomiRank Centrality reveals structural fragility of complex networks via node dominance","volume":"15","author":"Engsig","year":"2024","journal-title":"Nat. Commun."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1016\/0306-4573(89)90048-4","article-title":"Similarity measures in scientometric research: The Jaccard index vers us Salton\u2019s cosine formula","volume":"25","author":"Hamers","year":"1989","journal-title":"Inf. Process. Manag."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"7332","DOI":"10.1073\/pnas.0610245104","article-title":"Structure and tie strengths in mobile communication networks","volume":"104","author":"Onnela","year":"2007","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"P10011","DOI":"10.1088\/1742-5468\/2010\/10\/P10011","article-title":"Bridgeness: A local index on edge significance in maintaining global connectivity","volume":"2010","author":"Cheng","year":"2010","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Liu, Y., Tang, M., Zhou, T., and Do, Y. (2015). Improving the accuracy of the k-shell method by removing redundant links: From a perspective of spreading dynamics. Sci. Rep., 5.","DOI":"10.1038\/srep13172"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"056109","DOI":"10.1103\/PhysRevE.65.056109","article-title":"Attack vulnerability of complex networks","volume":"65","author":"Holme","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"123877","DOI":"10.1016\/j.physa.2019.123877","article-title":"Identifying significant edges via neighborhood information","volume":"548","author":"Zhao","year":"2020","journal-title":"Phys. Stat. Mech. Its Appl."},{"key":"ref_25","first-page":"1244","article-title":"Quantifying Importance of Edges in Networks","volume":"65","author":"Ouyang","year":"2018","journal-title":"IEEE Trans. Circuits Syst. II Express Briefs"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"098901","DOI":"10.1088\/1674-1056\/aca6d8","article-title":"Important edge identification in complex networks based on local and global features","volume":"32","author":"Song","year":"2023","journal-title":"Chin. Phys. B"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/0167-6377(89)90003-5","article-title":"Finding the most vital arcs in a network","volume":"8","author":"Ball","year":"1989","journal-title":"Oper. Res. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"7821","DOI":"10.1073\/pnas.122653799","article-title":"Community structure in social and biological networks","volume":"99","author":"Girvan","year":"2002","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1016\/j.knosys.2012.01.007","article-title":"A novel measure of edge centrality in social networks","volume":"30","author":"Ferrara","year":"2012","journal-title":"Knowl.-Based Syst."},{"key":"ref_30","unstructured":"Saito, K., Kimura, M., Ohara, K., and Motoda, H. (2016, January 22\u201326). Detecting critical links in complex network to maintain information flow\/reachability. Proceedings of the PRICAI 2016: Trends in Artificial Intelligence: 14th Pacific Rim International Conference on Artificial Intelligence, Phuket, Thailand. Proceedings 14."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"094102","DOI":"10.1103\/PhysRevLett.97.094102","article-title":"Characterizing the dynamical importance of network nodes and links","volume":"97","author":"Restrepo","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"016103","DOI":"10.1103\/PhysRevE.81.016103","article-title":"Finding communities in directed networks","volume":"81","author":"Kim","year":"2010","journal-title":"Phys. Rev. E\u2014Stat. Nonlinear, Soft Matter Phys."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"033115","DOI":"10.1063\/1.5081098","article-title":"Centrality-based identification of important edges in complex networks","volume":"29","author":"Lehnertz","year":"2019","journal-title":"Chaos: Interdiscip. J. Nonlinear Sci."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1137\/0201010","article-title":"Depth-first search and linear graph algorithms","volume":"1","author":"Tarjan","year":"1972","journal-title":"SIAM J. Comput."},{"key":"ref_35","unstructured":"(2024, July 02). United States Federal Aviation Administration, \u2018Air Traffic Control System Command Center\u2019, Available online: http:\/\/www.fly.faa.gov\/."},{"key":"ref_36","unstructured":"Barab\u00e1si, A., and P\u00f3sfai, M. (2016). Network Science, Cambridge University Press."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/S0378-8733(97)00016-6","article-title":"Exploring social structure using dynamic three-dimensional color images","volume":"20","author":"Freeman","year":"1998","journal-title":"Soc. Networks"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"e16962","DOI":"10.7554\/eLife.16962","article-title":"The CNS connectome of a tadpole larva of Ciona intestinalis (L.) highlights sidedness in the brain of a chordate sibling","volume":"5","author":"Ryan","year":"2016","journal-title":"eLife"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Ewing, R.M., Chu, P., Elisma, F., Li, H., Taylor, P., Climie, S., McBroom-Cerajewski, L., Robinson, M.D., O\u2019Connor, L., and Li, M. (2007). Large-scale mapping of human protein\u2013protein interactions by mass spectrometry. Mol. Syst. Biol., 3.","DOI":"10.1038\/msb4100134"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Adamic, L.A., and Glance, N. (2005, January 21\u201325). The political blogosphere and the 2004 US election: Divided they blog. Proceedings of the 3rd international workshop on Link discovery, Chicago, IL, USA.","DOI":"10.1145\/1134271.1134277"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"109521","DOI":"10.1016\/j.dib.2023.109521","article-title":"A Congressional Twitter network dataset quantifying pairwise probability of influence","volume":"50","author":"Fink","year":"2023","journal-title":"Data Brief"},{"key":"ref_42","unstructured":"Boyce, D.E., Chon, K.S., Ferris, M., Lee, Y.J., Lin, K., and Eash, R. (2025, March 23). Implementation and Evaluation of Combined Models of Urban Travel and Location on a Sketch Planning Network. Available online: https:\/\/trid.trb.org\/View\/268892."},{"key":"ref_43","unstructured":"Leskovec, J., and Mcauley, J. (2012). Learning to discover social circles in ego networks. Adv. Neural Inf. Process. Syst., 25, Available online: https:\/\/cs.stanford.edu\/people\/jure\/pubs\/circles-nips12.pdf."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"198701","DOI":"10.1103\/PhysRevLett.87.198701","article-title":"Efficient behavior of small-world networks","volume":"87","author":"Latora","year":"2001","journal-title":"Phys. Rev. Lett."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1016\/j.physa.2013.10.047","article-title":"Identifying and ranking influential spreaders in complex networks by neighborhood coreness","volume":"395","author":"Bae","year":"2014","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_46","first-page":"18","article-title":"On random graphs I","volume":"6","year":"1959","journal-title":"Publ. Math. Debr."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"760","DOI":"10.1016\/j.jtbi.2010.03.024","article-title":"Scaling behaviors of weighted food webs as energy transportation networks","volume":"264","author":"Zhang","year":"2010","journal-title":"J. Theor. Biol."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1016\/j.physa.2015.05.070","article-title":"Flow distances on open flow networks","volume":"437","author":"Guo","year":"2015","journal-title":"Phys. A Stat. Mech. Its Appl."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/4\/258\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:04:40Z","timestamp":1760029480000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/4\/258"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,28]]},"references-count":48,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["axioms14040258"],"URL":"https:\/\/doi.org\/10.3390\/axioms14040258","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,3,28]]}}}