From simple to complex networks: inherent structures, barriers and valleys in the context of spin glasses

TL;DR

This paper explores the energy landscape of spin glasses on various graphs, revealing statistical properties of inherent structures, energy barriers, and valleys, with implications for understanding complex systems.

Contribution

It introduces a comprehensive analysis of energy landscapes in spin glasses across different graph topologies using the lid algorithm, highlighting universal and topology-dependent features.

Findings

Inherent structures follow a lognormal distribution.

Energy barrier heights scale with system size.

Graph topology influences quantitative landscape features.

Abstract

Given discrete degrees of freedom (spins) on a graph interacting via an energy function, what can be said about the energy local minima and associated inherent structures? Using the lid algorithm in the context of a spin glass energy function, we investigate the properties of the energy landscape for a variety of graph topologies. First, we find that the multiplicity Ns of the inherent structures generically has a lognormal distribution. In addition, the large volume limit of ln<Ns>/<lnNs> differs from unity, except for the Sherrington-Kirkpatrick model. Second, we find simple scaling laws for the growth of the height of the energy barrier between the two degenerate ground states and the size of the associated valleys. For finite connectivity models, changing the topology of the underlying graph does not modify qualitatively the energy landscape, but at the quantitative level the models can differ substantially.