A measure of similarity between graph vertices
<|reference_start|>A measure of similarity between graph vertices: 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.<|reference_end|>
@article{blondel2004a,
    title={A measure of similarity between graph vertices},
    author={Vincent Blondel, Anahi Gajardo, Maureen Heymans, Pierre Senellart,
  Paul Van Dooren},
    journal={arXiv preprint arXiv:cs/0407061},
    year={2004},
    archivePrefix={arXiv},
    eprint={cs/0407061},
    primaryClass={cs.IR cond-mat.dis-nn cs.DM physics.data-an}
}