Scaling of directed dynamical small-world networks with random responses.

A dynamical model of small-world networks, with directed links which describe various correlations in social and natural phenomena, is presented. Random responses of sites to the input message are introduced to simulate real systems. The interplay of these ingredients results in the collective dynamical evolution of a spinlike variable S(t) of the whole network. The global average spreading length (s) and average spreading time (s) are found to scale as p(-alpha)ln(N with different exponents. Meanwhile, S(t) behaves in a duple scaling form for N>>N(*): S approximately f(p(-beta)q(gamma)t), where p and q are rewiring and external parameters, alpha, beta, and gamma are scaling exponents, and f(t) is a universal function. Possible applications of the model are discussed.