%0 Journal Article %1 Blondel04 %A Blondel, Vincent D. %A Gajardo, Anah\'ı %A Heymans, Maureen %A Senellart, Pierre %A Dooren, Paul Van %C Philadelphia, PA, USA %D 2004 %I Society for Industrial and Applied Mathematics %J SIAM Rev. %K LinearAlgebra graph similarity %N 4 %P 647--666 %R http://dx.doi.org/10.1137/S0036144502415960 %T A Measure of Similarity between Graph Vertices: Applications to Synonym Extraction and Web Searching %U http://portal.acm.org/citation.cfm?id=1035533.1035557 %V 46 %X We introduce a concept of similarity between vertices of directed graphs. Let GA and GB be two directed graphs with, respectively, nA and nB vertices. We define an nB times nA similarity matrix S whose real entry sij expresses how similar vertex j (in GA) is to vertex i (in GB): we say that sij is their similarity score. The similarity matrix can be obtained as the limit of the normalized even iterates of Sk+1 = BSkAT + BTSkA, where A and B are adjacency matrices of the graphs and S0 is a matrix whose entries are all equal to 1. In the special case where GA = GB = G, the matrix S is square and the score sij is the similarity score between the vertices i and j of G. We point out that Kleinberg's "hub and authority" method to identify web-pages 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.

@article{Blondel04, abstract = {We introduce a concept of {similarity} between vertices of directed graphs. Let GA and GB be two directed graphs with, respectively, nA and nB vertices. We define an nB times nA similarity matrix S whose real entry sij expresses how similar vertex j (in GA) is to vertex i (in GB): we say that sij is their similarity score. The similarity matrix can be obtained as the limit of the normalized even iterates of Sk+1 = BSkAT + BTSkA, where A and B are adjacency matrices of the graphs and S0 is a matrix whose entries are all equal to 1. In the special case where GA = GB = G, the matrix S is square and the score sij is the similarity score between the vertices i and j of G. We point out that Kleinberg's "hub and authority" method to identify web-pages 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.}, added-at = {2008-09-24T15:07:28.000+0200}, address = {Philadelphia, PA, USA}, author = {Blondel, Vincent D. and Gajardo, Anah\'{\i} and Heymans, Maureen and Senellart, Pierre and Dooren, Paul Van}, biburl = {https://www.bibsonomy.org/bibtex/2fbaef7a3057ff12e16dfd65c42fb0239/mkroell}, description = {A Measure of Similarity between Graph Vertices}, doi = {http://dx.doi.org/10.1137/S0036144502415960}, interhash = {b59d33c99477e70a646615cd0470f459}, intrahash = {fbaef7a3057ff12e16dfd65c42fb0239}, issn = {0036-1445}, journal = {SIAM Rev.}, keywords = {LinearAlgebra graph similarity}, number = 4, pages = {647--666}, publisher = {Society for Industrial and Applied Mathematics}, timestamp = {2008-09-24T15:07:28.000+0200}, title = {A Measure of Similarity between Graph Vertices: Applications to Synonym Extraction and Web Searching}, url = {http://portal.acm.org/citation.cfm?id=1035533.1035557}, volume = 46, year = 2004 }