Google matrix, dynamical attractors and Ulam networks

TL;DR

This paper investigates the properties of Google matrices derived from dynamical maps using the Ulam method, revealing scale-free networks and phase transitions affecting PageRank localization.

Contribution

It introduces a novel approach to analyze dynamical systems with the Google matrix framework, linking attractors to web-like network structures and PageRank behavior.

Findings

Ulam networks exhibit scale-free properties similar to the Web

PageRank localizes on dynamical attractors under certain parameters

Parameter variation can delocalize PageRank, reducing search efficiency

Abstract

We study the properties of the Google matrix generated by a coarse-grained Perron-Frobenius operator of the Chirikov typical map with dissipation. The finite size matrix approximant of this operator is constructed by the Ulam method. This method applied to the simple dynamical model creates the directed Ulam networks with approximate scale-free scaling and characteristics being rather similar to those of the World Wide Web. The simple dynamical attractors play here the role of popular web sites with a strong concentration of PageRank. A variation of the Google parameter $\alpha$ or other parameters of the dynamical map can drive the PageRank of the Google matrix to a delocalized phase with a strange attractor where the Google search becomes inefficient.