Small world of Ulam networks for chaotic Hamiltonian dynamics

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.