Thermodynamic stability of small-world oscillator networks: A case study of proteins

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.