# Geometrical features of time series provide new perspectives on collective fluctuations in driven disordered systems

## TL;DR

This paper introduces a novel approach using algebraic topology to analyze higher-order structures in time-series networks, revealing insights into collective fluctuations in driven disordered systems like ferromagnets.

## Contribution

It extends graph analysis of time series by incorporating algebraic topology to detect complex structures linked to collective fluctuations.

## Key findings

- High-order cliques correlate with collective fluctuations in hysteresis loops.
- Different stochastic processes are identified in different parts of the hysteresis cycle.
- Multifractal analysis confirms the presence of distinct fluctuation regimes.

## Abstract

Mapping time series onto graphs and the use of graph theory methods opens up the possibility to study the structure of the phase space manifolds underlying the fluctuations of a dynamical variable. Here, we propose to go beyond the standard graph measures and analyze the higher-order structures such as triangles, tetrahedra and higher-order cliques and their complexes occurring in the time-series networks, which are detectable by the algebraic topology methods. We apply the methodology to the signal of Barkhausen noise accompanying the domain-wall dynamics on the hysteresis loop of disordered ferromagnets driven by the external field. Our analysis demonstrates how the appearance of the complexes with cliques of a high order correlates to the enhanced collective fluctuations in the central part of the hysteresis loop, in contrast to the fractional Gaussian noise fluctuations at the beginning of the loop, which correspond to the graph of a simpler topology. The multifractal analysis of the corresponding segments of these time series confirms that we deal with different types of the stochastic process.