{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T14:37:45Z","timestamp":1770993465539,"version":"3.50.1"},"reference-count":10,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2006,10,11]],"date-time":"2006-10-11T00:00:00Z","timestamp":1160524800000},"content-version":"vor","delay-in-days":6798,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Networks"],"published-print":{"date-parts":[[1988,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The concept of vulnerability of a network, by which we mean the susceptibility of the network to attack, is very useful for the design of networks such as computer networks and communication networks. In this paper, a directed flow network, in which a nonnegative real number called edge capacity or capacity is associated with each edge is considered. If an edge in a network is destroyed, the value of maximum flow between two vertices in the network is decreased in general. If the decrease of the value of maximum flow by the destruction is large, the degree of influence of the edge on the vulnerability of the network is considered to be large. Two indices measuring the degree of influence of an edge on the vulnerability in the above sense are proposed and their properties are studied in this paper. Furthermore, the two indices are generalized to functions measuring the degree of influence of an edge on the vulnerability in a network.<\/jats:p>","DOI":"10.1002\/net.3230180109","type":"journal-article","created":{"date-parts":[[2007,5,11]],"date-time":"2007-05-11T23:20:05Z","timestamp":1178925605000},"page":"73-83","source":"Crossref","is-referenced-by-count":8,"title":["On a function for the vulnerability of a directed flow network"],"prefix":"10.1002","volume":"18","author":[{"given":"Masakazu","family":"Sengoku","sequence":"first","affiliation":[]},{"given":"Shoji","family":"Shinoda","sequence":"additional","affiliation":[]},{"given":"Reigo","family":"Yatsuboshi","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2006,10,11]]},"reference":[{"key":"e_1_2_1_2_2","volume-title":"Communication, Transmission, and Transportation Networks","author":"Frank F.","year":"1971"},{"issue":"4","key":"e_1_2_1_3_2","first-page":"352","article-title":"Axiomatic foundations of the theory of functions expressing the mediality of a point in a metric space","volume":"66","author":"Shinoda S.","year":"1983","journal-title":"Trans. ofIECE of Japan"},{"key":"e_1_2_1_4_2","volume-title":"Structural Model","author":"Harary F.","year":"1965"},{"issue":"11","key":"e_1_2_1_5_2","first-page":"745","article-title":"Intersection functions of shortest paths in a non\u2010directed graph","volume":"63","author":"Kishi G.","year":"1980","journal-title":"Trans. of IECE of Japan"},{"key":"e_1_2_1_6_2","unstructured":"Y.Kajitani The betweenness is a centrality function. Technical Report of IECE of Japan CAS84\u201019 (1984)."},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1287\/opre.18.3.497"},{"key":"e_1_2_1_8_2","doi-asserted-by":"publisher","DOI":"10.1515\/9781400875184"},{"key":"e_1_2_1_9_2","volume-title":"Finite Graphs and Networks: An Introduction with Applications","author":"Busacker R. G.","year":"1965"},{"key":"e_1_2_1_10_2","volume-title":"Network Flow, Transportation and Scheduling Theory and Algorithms","author":"Iri M.","year":"1969"},{"key":"e_1_2_1_11_2","doi-asserted-by":"publisher","DOI":"10.1145\/321694.321699"}],"container-title":["Networks"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fnet.3230180109","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/net.3230180109","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,21]],"date-time":"2023-10-21T20:16:55Z","timestamp":1697919415000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/net.3230180109"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1988,3]]},"references-count":10,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1988,3]]}},"alternative-id":["10.1002\/net.3230180109"],"URL":"https:\/\/doi.org\/10.1002\/net.3230180109","archive":["Portico"],"relation":{},"ISSN":["0028-3045","1097-0037"],"issn-type":[{"value":"0028-3045","type":"print"},{"value":"1097-0037","type":"electronic"}],"subject":[],"published":{"date-parts":[[1988,3]]}}}