Google matrix, dynamical attractors, and Ulam networks

Area of Science:

• Complex Systems
• Network Science
• Dynamical Systems

Background:

• The Google matrix, derived from the Perron-Frobenius operator, models complex networks.
• Dynamical systems with dissipation can exhibit properties analogous to real-world networks like the World Wide Web.

Purpose of the Study:

• To investigate the properties of a Google matrix generated from a coarse-grained Perron-Frobenius operator of a dissipative dynamical system.
• To explore the network characteristics and PageRank behavior of the resulting Ulam networks.

Main Methods:

• Construction of a finite-size matrix approximant using the Ulam method.
• Analysis of the generated directed Ulam networks for scale-free properties and PageRank distribution.
• Comparison of network characteristics with the World Wide Web.

Main Results:

• The Ulam networks exhibit approximate scale-free scaling and degree distributions.
• PageRank demonstrates a power-law decay, similar to the World Wide Web, with sensitivity to the Google parameter alpha.
• Dynamical attractors concentrate PageRank, acting like popular websites.

Conclusions:

• Dynamical systems can generate networks with web-like characteristics.
• Parameter variations in the dynamical map can lead to PageRank delocalization and inefficient search behavior.