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Graph fibrations, graph isomorphism, and PageRank

Published online by Cambridge University Press:  20 July 2006

Paolo Boldi ,
Violetta Lonati ,
Massimo Santini  and
Sebastiano Vigna
Paolo Boldi
Affiliation:
Dipartimento di Scienze dell'Informazione, Università degli Studi di Milano, Italy; santini@dsi.unimi.it
Violetta Lonati
Affiliation:
Dipartimento di Scienze dell'Informazione, Università degli Studi di Milano, Italy; santini@dsi.unimi.it
Massimo Santini
Affiliation:
Dipartimento di Scienze dell'Informazione, Università degli Studi di Milano, Italy; santini@dsi.unimi.it
Sebastiano Vigna
Affiliation:
Dipartimento di Scienze dell'Informazione, Università degli Studi di Milano, Italy; santini@dsi.unimi.it
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Abstract

PageRank is a ranking method that assigns scores to web pages using the limit distribution of a random walk on the web graph. A fibration of graphs is a morphism that is a local isomorphism of in-neighbourhoods, much in the same way a covering projection is a local isomorphism of neighbourhoods. We show that a deep connection relates fibrations and Markov chains with restart, a particular kind of Markov chains that include the PageRank one as a special case. This fact provides constraints on the values that PageRank can assume. Using our results, we show that a recently defined class of graphs that admit a polynomial-time isomorphism algorithm based on the computation of PageRank is really a subclass of fibration-prime graphs, which possess simple, entirely discrete polynomial-time isomorphism algorithms based on classical techniques for graph isomorphism. We discuss efficiency issues in the implementation of such algorithms for the particular case of web graphs, in which O(n) space occupancy (where n is the number of nodes) may be acceptable, but O(m) is not (where m is the number of arcs).

Keywords

Graph fibrationsPageRankMarkov chain with restart.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 40 , Issue 2: Alberto Bertoni: Climbing summits , April 2006 , pp. 227 - 253
Copyright
© EDP Sciences, 2006

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