  In this paper, we offer a competing dynamic analysis of the one-dimensional
Ising model built on the small-world network (SWN). Adding-type SWNs are
investigated in detail using a simplified Hamiltonian of mean-field nature, and
the result of rewiring-type is given because of the similarities of these two
typical networks. We study the dynamical processes with competing Glauber
mechanism and Kawasaki mechanism. The Glauber-type single-spin transition
mechanism with probability p simulates the contact of the system with a heat
bath and the Kawasaki-type dynamics with probability 1-p simulates an external
energy flux. By studying the phase diagram obtained in the present work, we can
realize some dynamical properties influenced by the small-world effect.
