Cavity approach for real variables on diluted graphs and application to synchronization in small-world lattices

TL;DR

This paper extends the cavity method to real variables on small-world graphs, revealing how such architectures enhance synchronization in XY spin systems and exploring replica symmetry breaking phenomena.

Contribution

It introduces a cavity approach for real-valued variables on diluted graphs and applies it to analyze synchronization and spin-glass phases in small-world networks.

Findings

Small-world architectures significantly expand synchronization regions.

Population dynamics and Monte Carlo simulations show excellent agreement.

Replica symmetry breaking occurs in the spin-glass phase.

Abstract

We study XY spin systems on small world lattices for a variety of graph structures, e.g. Poisson and scale-free, superimposed upon a one dimensional chain. In order to solve this model we extend the cavity method in the one pure-state approximation to deal with real-valued dynamical variables. We find that small-world architectures significantly enlarge the region in parameter space where synchronization occurs. We contrast the results of population dynamics performed on a truncated set of cavity fields with Monte Carlo simulations and find excellent agreement. Further, we investigate the appearance of replica symmetry breaking in the spin-glass phase by numerically analyzing the proliferation of pure states in the message passing equations.