Boolean decision problems with competing interactions on scale-free networks: Critical thermodynamics

TL;DR

This paper investigates the critical behavior of Boolean variables on scale-free networks with competing interactions, revealing phase transitions, stability, and universality classes in spin-glass systems through analytical and simulation methods.

Contribution

It provides the first detailed phase diagram and universality analysis for Ising spin glasses on scale-free networks with varying decay exponents.

Findings

Finite-temperature spin-glass transition occurs for decay exponent > 3.

Spin-glass phase remains stable at all temperatures for decay exponent ≤ 3.

Universality class matches mean-field Sherrington-Kirkpatrick model when decay exponent > 4.

Abstract

We study the critical behavior of Boolean variables on scale-free networks with competing interactions (Ising spin glasses). Our analytical results for the disorder-network-decay-exponent phase diagram are verified using Monte Carlo simulations. When the probability of positive (ferromagnetic) and negative (antiferromagnetic) interactions is the same, the system undergoes a finite-temperature spin-glass transition if the exponent that describes the decay of the interaction degree in the scale-free graph is strictly larger than 3. However, when the exponent is equal to or less than 3, a spin-glass phase is stable for all temperatures. The robustness of both the ferromagnetic and spin-glass phases suggests that Boolean decision problems on scale-free networks are quite stable to local perturbations. Finally, we show that for a given decay exponent spin glasses on scale-free networks seem to obey universality. Furthermore, when the decay exponent of the interaction degree is larger than 4 in the spin-glass sector, the universality class is the same as for the mean-field Sherrington-Kirkpatrick Ising spin glass.