Similarity matrices for pairs of graphs

We introduce a concept of similarity between vertices of directed graphs. Let G(A) and G(B) be two directed graphs with respectively n(A) and n(B) vertices. We define a n(A) x n(B) similarity matrix S whose real entry s(ij) expresses how similar vertex i (in G(A)) is to vertex j (in G(B)) we say that s(ij) is their similarity score. In the special case where G(A) = G(B) = G, the score s(ij) 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 Klemberg, 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