Authors 
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Document type 
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Article de périodique (Journal article) – Article de recherche

Abstract 
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We introduce a concept of similarity between vertices of directed graphs. Let CA and G(B) be two directed graphs with, respectively, n(A) and n(B) vertices. We define an n(B) x n(A) similarity matrix S whose real entry s(ij) expresses how similar vertex j (in G(A)) is to vertex i (in G(B)): we say that s(ij) is their similarity score. The similarity matrix can be obtained as the limit of the normalized even iterates of Sk+1 = BS(k)A(T) + B(T)S(k)A, where A and B are adjacency matrices of the graphs and So is a matrix whose entries are all equal to 1. In the special case where G(A) = G(B) = G, the matrix S is square and the score s(ij) is the similarity score between the vertices i and j of G. We point out that Klemberg's "hub and authority" method to identify webpages relevant to a given query can be viewed as a special case of our definition in the case where one of the graphs has two vertices and a unique directed edge between them. In analogy to Kleinberg, we show that our similarity scores are given by the components of a dominant eigenvector of a nonnegative matrix. Potential applications of our similarity concept are numerous. We illustrate an application for the automatic extraction of synonyms in a monolingual dictionary. 
Access type 
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Accès restreint 
Publication date 
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2004 
Language 
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Anglais 
Journal information 
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"SIAM Review"  Vol. 46, no. 4, p. 647666 (2004) 
Peer reviewed 
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yes 
Publisher 
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Siam Publications (Philadelphia)

issn 
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00361445 
eissn 
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10957200 
Publication status 
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Publié 
Affiliation 
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UCL
 FSA/INMA  Département d'ingénierie mathématique

Keywords 
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algorithms ; graph algorithms ; graph theory ; eigenvalues of graphs

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