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 p1/N , S approximately N ; (2) a stable phase: when p1/N , S approximately p;{-1} , i.e., S/N approximately 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 approximately 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 approximately E_{cr};{-1} , regardless of the model details.