A Measure of Similarity between Graph Vertices: Applications to Synonym Extraction and Web Searching

Abstract

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 B × 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 S(k +1) = BS(k)A T +B T 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 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 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.

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