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 go beyond the standard graph measures and analyze the higher-order structures such as triangles, tetrahedra and higher-order cliques and their complexes present in the time-series networks, which are detectable by algebraic topology methods. We investigate the Barkhausen noise signal which accompanies domain-wall dynamics during magnetization reversal in weakly disordered ferromagnets by a slow increase of the external field along the hysteresis loop. 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, where domain-wall depinning occurs. In contrast, the fractional Gaussian noise fluctuations at the beginning of the loop correspond to the graph of a simpler topology. The determined topology measures serve as geometrical indicators of changing dynamical regimes along the hysteresis loop. The multifractal analysis of the corresponding segments of the signal confirms that we deal with different types of stochastic processes.
PACS Nos 12.60.Jv; 12.10.Dm; 98.80.Cq; 11.30.Hv