PageRank equation and localization in the WWW

¹  Department of Physics, SLACS-CNR University of Cagliari - S.P. Monserrato-Sestu Km 0.700, 09042 Cagliari, Italy, EU
²  Linkalab, Complex Systems Computational Laboratory - 09100 Cagliari Italy, EU
³  Center for Complex Networks and Systems Research, School of Informatics and Computing, Indiana University Bloomington, IN, USA
⁴  Centre SMC CNR-INFM, Department of Physics, University “Sapienza” - P.le Moro 5, 00185 Rome, Italy, EU
⁵  Theoretical Physics Division, Rudjer Boskovic Institute - P.O. Box 180, HR-10002 Zagreb, Croatia
⁶  Centre SOFT INFM-CNR, Department of Physics, University “Sapienza” - P.le Moro 5, 00185 Rome, Italy, EU
⁷  Institute for Complex Systems ISC-CNR - Via dei Taurini 19, 00185 Rome, Italy, EU
⁸  Yahoo! Research - Avinguda Diagonal 177, 8th floor, 08018 Barcelona, Spain, EU

Corresponding author: nicolaperra@gmail.com

Received: 17 August 2009
Accepted: 23 October 2009

Abstract

We show that the PageRank in a network can be represented as the solution of a differential equation discretized over a directed graph. By exploiting a formal relationship with the time-independent Schrödinger equation it is possible to interpret hub formation and related phenomena as a wave-like localization process in the presence of disorder and trapping potentials. The result opens new perspectives in the physics of networks with interdisciplinary connections and opens the way to the employment of various mathematical techniques to the analysis of self-organization in structured systems. Applications are envisaged in the World-Wide Web, traffic, social and biological networks.