  We study vibrational thermodynamic stability of small-world oscillator
networks, by relating the average mean-square displacement $S$ of oscillators
to the eigenvalue spectrum of the Laplacian matrix of networks. We show that
the cross-links suppress $S$ effectively and there exist two phases on the
small-world networks: 1) an unstable phase: when $p\ll1/N$, $S\sim N$; 2) a
stable phase: when $p\gg1/N$, $S\sim p^{-1}$, \emph{i.e.}, $S/N\sim
E_{cr}^{-1}$. Here, $p$ is the parameter of small-world, $N$ is the number of
oscillators, and $E_{cr}=pN$ is the number of cross-links. The results are
exemplified by various real protein structures that follow the same scaling
behavior $S/N\sim E_{cr}^{-1}$ of the stable phase. We also show that it is the
"small-world" property that plays the key role in the thermodynamic stability
and is responsible for the universal scaling $S/N\sim E_{cr}^{-1}$, regardless
of the model details.
