Similarity Matrices for Pairs of Graphs

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 nA × nB similarity matrix S whose real entry sij expresses how similar vertex i (in GA) is to vertex j (in GB) : we say that sij is their similarity score. In the special case where GA = GB = G, the score sij is the similarity score between the vertices i and j of G and the square similarity matrix S is the self-similarity matrix of the graph 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 vector of a non-negative matrix and we propose a simple iterative method to compute them.