On metastable configurations of small-world networks

TL;DR

This paper analyzes the number and properties of metastable states in small-world networks modeled with Ising spins, revealing entropy jumps at structural crossovers using advanced transfer-matrix methods.

Contribution

It introduces a novel calculation of metastable configurations in small-world networks with sparse Poisson graphs, highlighting entropy behavior at structural transitions.

Findings

Entropy of metastable states exhibits a jump at the small-world to Poisson graph crossover.

Ground state degeneracy is characterized within the network models.

Differences in entropy between metastable and all configurations are quantified for various couplings.

Abstract

We calculate the number of metastable configurations of Ising small-world networks which are constructed upon superimposing sparse Poisson random graphs onto a one-dimensional chain. Our solution is based on replicated transfer-matrix techniques. We examine the denegeracy of the ground state and we find a jump in the entropy of metastable configurations exactly at the crossover between the small-world and the Poisson random graph structures. We also examine the difference in entropy between metastable and all possible configurations, for both ferromagnetic and bond-disordered long-range couplings.