Heterogeneous nucleation in finite size adaptive dynamical networks

TL;DR

This paper investigates how adaptive, heterogeneous networks of oscillators undergo distinct first-order phase transitions, revealing phenomena similar to nucleation, and develops a mean-field theory to understand these complex dynamics.

Contribution

It introduces a theoretical framework for analyzing phase transitions in finite-size adaptive networks with heterogeneity, highlighting the role of defects and multicluster states.

Findings

Identifies two types of first-order phase transitions depending on defect nature.

Shows the emergence of multicluster states influencing transition character.

Develops a mean-field model capturing the interplay of adaptivity and heterogeneity.

Abstract

Phase transitions in equilibrium and nonequilibrium systems play a major role in the natural sciences. In dynamical networks, phase transitions organize qualitative changes in the collective behavior of coupled dynamical units. Adaptive dynamical networks feature a connectivity structure that changes over time, co-evolving with the nodes' dynamical state. In this Letter, we show the emergence of two distinct first-order nonequilibrium phase transitions in a finite size adaptive network of heterogeneous phase oscillators. Depending on the nature of defects in the internal frequency distribution, we observe either an abrupt single-step transition to full synchronization or a more gradual multi-step transition. This observation has a striking resemblance to heterogeneous nucleation. We develop a mean-field approach to study the interplay between adaptivity and nodal heterogeneity and describe the dynamics of multicluster states and their role in determining the character of the phase transition. Our work provides a theoretical framework for studying the interplay between adaptivity and nodal heterogeneity.