A measure of similarity between graph vertices

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 a nB × 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 S(k+1) = BS(k)A +B S(k)A where A and B are adjacency matrices of the graphs and S(0) is a matrix whose entries are all equal to one. 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 non-negative matrix. Potential applications of our similarity concept are numerous. We illustrate an application for the automatic extraction of synonyms in a monolingual dictionary.