Multiscaling in the YX model of networks

TL;DR

This paper explores a Hamiltonian network model inspired by the XY model, revealing complex scaling behaviors and phase-like regimes without finite-temperature criticality, with network structures varying across energy levels.

Contribution

It introduces a novel network Hamiltonian model conserving spins and sampling network topologies, analyzing its scaling behavior and structural phases.

Findings

No finite-temperature critical behavior observed.

Three distinct structural regimes identified at finite sizes.

Different network properties approach T=0-values with varying exponents.

Abstract

We investigate a Hamiltonian model of networks. The model is a mirror formulation of the XY model (hence the name) -- instead letting the XY spins vary, keeping the coupling topology static, we keep the spins conserved and sample different underlying networks. Our numerical simulations show complex scaling behaviors, but no finite-temperature critical behavior. The ground state and low-order excitations for sparse, finite graphs is a fragmented set of isolated network clusters. Configurations of higher energy are typically more connected. The connected networks of lowest energy are stretched out giving the network large average distances. For the finite sizes we investigate there are three regions -- a low-energy regime of fragmented networks, and intermediate regime of stretched-out networks, and a high-energy regime of compact, disordered topologies. Scaling up the system size, the borders between these regimes approach zero temperature algebraically, but different network structural quantities approach their T=0-values with different exponents.