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A measure of similarity between graph vertices

Vincent Blondel·Anahi Gajardo·Maureen Heymans·Pierre Senellart·Paul Van Dooren
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Abstract

We introduce a concept of similarity between vertices of directed graphs. Let G_A and G_B be two directed graphs. We define a similarity matrix whose (i, j)-th real entry expresses how similar vertex j (in G_A) is to vertex i (in G_B. The similarity matrix can be obtained as the limit of the normalized even iterates of a linear transformation. In the special case where G_A=G_B=G, the matrix is square and the (i, j)-th entry 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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Vincent Blondel, Anahi Gajardo, Maureen Heymans, Pierre Senellart, Paul Van Dooren. 2004-07-28. A measure of similarity between graph vertices. https://arxiv.org/abs/cs/0407061

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