Closeness Centrality Extended to Unconnected Graphs: the Harmonic Centrality Index

AbstractSocial network analysis is a rapid expanding interdisciplinary field,growingfromworkofsociologists,physicists,historians,mathematicians,political scientists, etc. Some methods have been commonly accepted inspite of defects, perhaps because of the rareness of synthetic work like(Freeman, 1978; Faust & Wasserman, 1992). In this article, we proposeanalternativeindexofclosenesscentralitydefinedonundirectednetworks.We show that results from its computation on real cases are identical tothoseoftheclosenesscentralityindex,withsamecomputationalcomplex-ityandwegivesomeinterpretations. Animportantpropertyisitsuseinthecaseofunconnectednetworks. 1 Introduction The study of centrality is one of the most popular subject in the analysisof social networks. Determining the role of an individual within a society, itsinfluence or the flows of information on which he can intervene are examplesofapplicationsofcentralityindices. Theyaredefinedatanactor-levelandareexpectedtocompareandbetterunderstandrolesofeachindividualinthenet-work. Inaddition,agraph-levelindexcalledcentralization(i.e. howmuchtheindex value of the most central node is bigger than the others) is defined foreachexistingindex.Each index provides a way to highlight properties of individuals, depen-dentlyonitsdefinition. Forexample,degreecentralityattributeshighmeasureto an individual having great influence on its neighbors. Closeness centralityhighlights the players who will be able to contact easily all other members ofthenetwork. Betweennesscentralitygiveshighestvaluestoindividualsthroughwhominformationismorelikelytopass.1

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