Dynamical Phase Transitions in Non-equilibrium Networks

TL;DR

This paper introduces a minimal model for classical nonequilibrium networks that exhibits dynamical phase transitions, characterized by critical scaling and divergence of network degree at finite times, bridging empirical observations and theoretical understanding.

Contribution

It provides the first theoretical framework demonstrating how nonlinear interactions in classical networks lead to dynamical phase transitions and critical scaling behaviors.

Findings

Network degree diverges at a finite critical time.

Degree distributions and clustering coefficients show critical scaling.

Universal hyperbolic scaling law at the transition.

Abstract

Dynamical phase transitions (DPTs) characterize critical changes in system behavior occurring at finite times, providing a lens to study nonequilibrium phenomena beyond conventional equilibrium physics. While extensively studied in quantum systems, DPTs have remained largely unexplored in classical settings. Recent experiments on complex systems, from social networks to financial markets, have revealed abrupt dynamical changes analogous to quantum DPTs, motivating the search for a theoretical understanding. Here, we present a minimal model for nonequilibrium networks, demonstrating that nonlinear interactions among network edges naturally give rise to DPTs. Specifically, we show that network degree diverges at a finite critical time, following a universal hyperbolic scaling, consistent with empirical observations. Our analytical results predict that key network properties, including degree distributions and clustering coefficients, exhibit critical scaling as criticality approaches. These findings establish a theoretical foundation for understanding emergent nonequilibrium criticality across diverse complex systems.