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

Abstract

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□1/N, S∼N; (2) a stable phase: when p□1/N, S∼ p-1, i.e., S/N∼ E cr -1. Here, p is the parameter of small-world, N is the number of oscillators, and Ecr =pN is the number of cross-links. The results are exemplified by various real protein structures that follow the same scaling behavior S/N∼ 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∼ E cr -1, regardless of the model details. © 2009 The American Physical Society.