  A dynamical model of small-world network, with directed links which describe
various correlations in social and natural phenomena, is presented. Random
responses of every site to the imput message are introduced to simulate real
systems. The interplay of these ingredients results in collective dynamical
evolution of a spin-like variable S(t) of the whole network. In the present
model, global average spreading length \langel L >_s and average spreading time
<T >_s are found to scale as p^-\alpha ln N with different exponents.
Meanwhile, S behaves in a duple scaling form for N>>N^*: S ~ f(p^-\beta
q^\gamma t'_sc), where p and q are rewiring and external parameters, \alpha,
\beta, \gamma and f(t'_sc) are scaling exponents and universal functions,
respectively. Possible applications of the model are discussed.
