Complex networks and glassy dynamics: walks in the energy landscape

TL;DR

This paper introduces a mathematical framework using complex network theory to analyze glassy system dynamics as random walks on energy landscape networks, revealing how topology influences glass transition properties.

Contribution

It extends mean-field models by incorporating network topology and energy-degree relationships, providing new insights into glassy phase transitions and dynamics.

Findings

Glassy phase existence depends on network topology and energy-degree relationship.

Degree correlations and transition rate details affect complex properties but not the transition temperature.

Low-temperature phase has two regions with different trapping time scaling.

Abstract

We present a simple mathematical framework for the description of the dynamics of glassy systems in terms of a random walk in a complex energy landscape pictured as a network of minima. We show how to use the tools developed for the study of dynamical processes on complex networks, in order to go beyond mean-field models that consider that all minima are connected to each other. We consider several possibilities for the transition rates between minima, and show that in all cases the existence of a glassy phase depends on a delicate interplay between the network's topology and the relationship between energy and degree of a minimum. Interestingly, the network's degree correlations and the details of the transition rates do not play any role in the existence (nor in the value) of the transition temperature, but have an impact only on more involved properties. For Glauber or Metropolis rates in particular, we find that the low-temperature phase can be further divided into two regions with different scaling properties of the average trapping time. Overall, our results rationalize and link the empirical findings about correlations between the energy of the minima and their degree, and should stimulate further investigations on this issue.