  We study the collective behavior of an Ising system on a small-world network
with the interaction $J(r) \propto r^{-\alpha}$, where $r$ represents the
Euclidean distance between two nodes. In the case of $\alpha = 0$ corresponding
to the uniform interaction, the system is known to possess a phase transition
of the mean-field nature, while the system with the short-range interaction
$(\alpha\to\infty)$ does not exhibit long-range order at any finite
temperature. Monte Carlo simulations are performed at various values of
$\alpha$, and the critical value $\alpha_c$ beyond which the long-range order
does not emerge is estimated to be zero. Thus concluded is the absence of a
phase transition in the system with the algebraically decaying interaction
$r^{-\alpha}$ for any nonzero positive value of $\alpha$.
