  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.
