Analyzing long-term correlated stochastic processes by means of recurrence networks: Potentials and pitfalls

Long-range correlated processes are ubiquitous, ranging from climate variables to financial time series. One paradigmatic example for such processes is fractional Brownian motion (fBm). In this work, we highlight the potentials and conceptual as well as practical limitations when applying the recently proposed recurrence network (RN) approach to fBm and related stochastic processes. In particular, we demonstrate that the results of a previous application of RN analysis to fBm (Liu \textit{et al.,} Phys. Rev. E \textbf{89}, 032814 (2014)) are mainly due to an inappropriate treatment disregarding the intrinsic non-stationarity of such processes. Complementarily, we analyze some RN properties of the closely related stationary fractional Gaussian noise (fGn) processes and find that the resulting network properties are well-defined and behave as one would expect from basic conceptual considerations. Our results demonstrate that RN analysis can indeed provide meaningful results for stationary stochastic processes, given a proper selection of its intrinsic methodological parameters, whereas it is prone to fail to uniquely retrieve RN properties for non-stationary stochastic processes like fBm.

💡 Research Summary

The paper critically examines the applicability of recurrence network (RN) analysis to long‑range correlated stochastic processes, focusing on fractional Brownian motion (fBm) and its stationary counterpart, fractional Gaussian noise (fGn). RN methodology constructs a graph from a time series by embedding the data in a reconstructed phase space (using delay vectors of dimension m and lag τ) and connecting points whose mutual distance is smaller than a prescribed threshold ε. This approach assumes that the underlying system is statistically stationary and that the embedding parameters faithfully capture the geometry of the attractor.

For fBm, which is intrinsically non‑stationary (its variance grows as t^{2H}), these assumptions break down. The authors demonstrate that neither the embedding dimension m nor the delay τ can be meaningfully defined for fBm. The fractal dimension of a stochastic trajectory diverges (or, in the graph‑based sense, equals 2 – H), implying that an infinite embedding dimension would be required to avoid projection artefacts. In practice, any finite m produces spurious results that depend strongly on the length N of the time series. Numerical experiments show that RN metrics such as average path length and clustering coefficient vary systematically with N even when the Hurst exponent H is held constant. Moreover, the autocorrelation function of fBm decays extremely slowly, rendering conventional methods for selecting τ (e.g., first zero‑crossing of the ACF or mutual information) ineffective. Consequently, the RN characteristics reported in earlier work (Liu et al., Phys. Rev. E 89, 032814, 2014) are largely artefacts of an inappropriate treatment of non‑stationarity rather than genuine signatures of the underlying dynamics.

In contrast, fGn—obtained by differencing fBm—is a stationary process with well‑defined autocorrelation decay that follows a power law determined by H. For fGn, standard embedding techniques (false nearest neighbours for m, first ACF zero or mutual information for τ) work reliably, and RN properties become robust with respect to series length. The authors generate fGn realizations across a wide range of H values (0.05 to 0.95) and construct RNs at fixed connection density (or fixed ε). They find that as H increases, the resulting networks exhibit systematic changes: average degree, clustering coefficient, and characteristic path length vary in a predictable manner that reflects the strength of long‑range correlations. Importantly, once m is chosen sufficiently large (e.g., m ≥ 4), the RN metrics converge and no longer depend on the specific realization or sample size, confirming that RN analysis can faithfully capture the geometric structure of stationary long‑range correlated processes.

The paper concludes with two practical recommendations. First, non‑stationary data should be transformed to a stationary form (e.g., by differencing or detrending) before applying RN methods. Second, researchers must verify that RN results are insensitive to the choice of ε, m, τ, and the length of the data, ideally by performing systematic sensitivity analyses. When these precautions are taken, RN analysis provides a powerful tool for quantifying the geometry of stationary stochastic processes such as fGn, but it is prone to produce misleading, parameter‑dependent artefacts when applied directly to non‑stationary processes like fBm.