A measure of similarity between graph vertices
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 webpages 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 nonnegative matrix. Potential applications of our similarity concept are numerous. We illustrate an application for the automatic extraction of synonyms in a monolingual dictionary.
 Publication:

arXiv eprints
 Pub Date:
 July 2004
 DOI:
 10.48550/arXiv.cs/0407061
 arXiv:
 arXiv:cs/0407061
 Bibcode:
 2004cs........7061B
 Keywords:

 Computer Science  Information Retrieval;
 Computer Science  Discrete Mathematics;
 Condensed Matter  Disordered Systems and Neural Networks;
 Physics  Data Analysis;
 Statistics and Probability