Complex ordering in spin networks: Critical role of adaptation rate for dynamically evolving interactions

TL;DR

This paper investigates how the rate of adaptive interaction changes influences the emergence of order, frustration, and complex energy landscapes in networks of Ising spins, revealing critical thresholds for different behaviors.

Contribution

It demonstrates that the adaptation rate of interactions critically determines the system's ordering and frustration, highlighting a phase transition driven by learning speed.

Findings

Fast adaptation leads to balanced, smooth energy landscapes.

Slow adaptation results in frustrated, rugged landscapes.

Small changes in adaptation rate cause qualitative shifts in system behavior.

Abstract

Many complex systems can be represented as networks of dynamical elements whose states evolve in response to interactions with neighboring elements, noise and external stimuli. The collective behavior of such systems can exhibit remarkable ordering phenomena such as chimera order corresponding to coexistence of ordered and disordered regions. Often, the interactions in such systems can also evolve over time responding to changes in the dynamical states of the elements. Link adaptation inspired by Hebbian learning, the dominant paradigm for neuronal plasticity, has been earlier shown to result in structural balance by removing any initial frustration in a system that arises through conflicting interactions. Here we show that the rate of the adaptive dynamics for the interactions is crucial in deciding the emergence of different ordering behavior (including chimera) and frustration in networks of Ising spins. In particular, we observe that small changes in the link adaptation rate about a critical value result in the system exhibiting radically different energy landscapes, viz., smooth landscape corresponding to balanced systems seen for fast learning, and rugged landscapes corresponding to frustrated systems seen for slow learning.