Delay-Controlled Heterogeneous Nucleation in Adaptive Dynamical Networks

TL;DR

This paper investigates how delay-induced heterogeneous nucleation influences phase transitions and synchronization in adaptive dynamical networks, providing analytical conditions and a mean-field framework.

Contribution

It introduces two forms of heterogeneous nucleation in delay-coupled adaptive networks and develops a mean-field model with analytical bounds for cluster states.

Findings

Delay and frequency distribution shape the nucleation type.

Analytical bounds match numerical simulations.

Extended analysis includes distributed delays.

Abstract

Phase transitions constitute fundamental mechanisms underlying abrupt or qualitative changes in the collective dynamics of interacting units across a wide range of natural and engineered systems. In dynamical networks, such transitions lead to significant reorganization in the coordinated behavior of coupled elements. In adaptive dynamical networks, the connectivity evolves dynamically in response to the states of the nodes, resulting in a coevolution of structure and dynamics. In this work, we report two distinct forms of heterogeneous nucleation that give rise to single-step and multi-step phase transitions toward global synchronization in finite-size adaptive networks with connection delays. We demonstrate that the nature of the nucleation transition is governed by both the presence and magnitude of the delay, as well as the class of natural frequency distribution. Using a collective coordinate framework, we develop a mean-field description of cluster dynamics and derive an analytical upper bound condition for the existence of two-cluster states, which shows excellent agreement with numerical simulations. Furthermore, we extend the analysis to systems with distributed delays and obtain corresponding analytical conditions. Our results provide a theoretical framework for understanding synchronization transitions in adaptive networks with time-delayed interactions.