Small world of Ulam networks for chaotic Hamiltonian dynamics

TL;DR

This paper demonstrates that Ulam networks derived from chaotic Hamiltonian maps exhibit small world properties, with the Erdős number growing logarithmically or algebraically depending on the dynamics, linking network structure to chaos characteristics.

Contribution

It reveals that Ulam networks for symplectic maps are small world networks and connects their properties to the underlying chaotic or stable dynamics.

Findings

Erdős number grows logarithmically with network size in strong chaos.

Presence of stability islands causes algebraic growth of Erdős number.

Erdős number and Perron-Frobenius relaxation times behave differently.

Abstract

We show that the Ulam method applied to dynamical symplectic maps generates Ulam networks which belong to the class of small world networks appearing for social networks of people, actors, power grids, biological networks and Facebook. We analyze the small world properties of Ulam networks on examples of the Chirikov standard map and the Arnold cat map showing that the number of degrees of separation, or the Erd\"os number, grows logarithmically with the network size for the regime of strong chaos. This growth is related to the Lyapunov instability of chaotic dynamics. The presence of stability islands leads to an algebraic growth of the Erd\"os number with the network size. We also compare the time scales related with the Erd\"os number and the relaxation times of the Perron-Frobenius operator showing that they have a different behavior.