Coordination sequences and information spreading in small-world networks

We study the spread of information in small-world networks generated from different d-dimensional regular lattices, with d=1, 2, and 3. With this purpose, we analyze by numerical simulations the behavior of the coordination sequence, e.g., the average number of sites C(n) that can be reached from a given node of the network in n steps along its bonds. For sufficiently large networks, we find an asymptotic behavior C(n)∼ρn, with a constant ρ that depends on the network dimension d and on the rewiring probability p (which measures the disorder strength of a given network). A simple model of information spreading in these networks is studied, assuming that only a fraction q of the network sites are active. The number of active nodes reached in n steps has an asymptotic form λn, λ being a constant that depends on p and q, as well as on the dimension d of the underlying lattice. The information spreading presents two different regimes depending on the value of λ: For λ>1 the information propagates along the whole system, and for λ<1 the spreading is damped and the information remains confined in a limited region of the network. We discuss the connection of these results with site percolation in small-world networks