Google matrix and Ulam networks of intermittency maps

TL;DR

This paper investigates the spectral properties of the Google matrix derived from Ulam networks of intermittency maps, revealing how PageRank behavior depends on map parameters and can become delocalized, affecting search efficiency.

Contribution

It introduces a novel analysis of the Google matrix for Ulam networks of intermittency maps, linking spectral properties to PageRank localization and delocalization phenomena.

Findings

PageRank follows a power law decay with map-dependent exponent

Under certain conditions, PageRank becomes completely delocalized

The spectral analysis reveals how map parameters influence eigenvalues and eigenvectors

Abstract

We study the properties of the Google matrix of an Ulam network generated by intermittency maps. This network is created by the Ulam method which gives a matrix approximant for the Perron-Frobenius operator of dynamical map. The spectral properties of eigenvalues and eigenvectors of this matrix are analyzed. We show that the PageRank of the system is characterized by a power law decay with the exponent $\beta$ dependent on map parameters and the Google damping factor $\alpha$. Under certain conditions the PageRank is completely delocalized so that the Google search in such a situation becomes inefficient.