  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.
