Uniform framework for the recurrence-network analysis of chaotic time series

Abstract

We propose a general method for the construction and analysis of unweighted ε-recurrence networks from chaotic time series. The selection of the critical threshold ε_{c} in our scheme is done empirically and we show that its value is closely linked to the embedding dimension M. In fact, we are able to identify a small critical range Δε numerically that is approximately the same for the random and several standard chaotic time series for a fixed M. This provides us a uniform framework for the nonsubjective comparison of the statistical measures of the recurrence networks constructed from various chaotic attractors. We explicitly show that the degree distribution of the recurrence network constructed by our scheme is characteristic to the structure of the attractor and display statistical scale invariance with respect to increase in the number of nodes N. We also present two practical applications of the scheme, detection of transition between two dynamical regimes in a time-delayed system and identification of the dimensionality of the underlying system from real-world data with a limited number of points through recurrence network measures. The merits, limitations, and the potential applications of the proposed method are also highlighted.

Quantifying information loss on chaotic attractors through recurrence networks

“…A critical range of threshold ∆ǫ is selected for the construction of the network whose minimum is chosen with the standard condition that there exists a single giant component in the resulting RN. We have shown that [37] this critical range is approximately identical for several chaotic systems and white noise and de-pends only on the embedding dimension M. The threshold values used here for the construction of the network are ǫ = 0.06, 0.10, 0.14 and 0.18 for M varying from 2 to 5 respectively. The scheme has been effectively employed to study the influence of noise on the structure of chaotic attractors [38] and for the analysis of light curves from a prominent black hole system [39].…”

“…The choice of the parameter ǫ is crucial for the construction of the RN to ensure that the resulting network is a proper representation of the embedded attractor. We have recently proposed a scheme [37] for the construction of RN where the value of ǫ is automated for a given embedding dimension M.…”

Recurrence network measures for hypothesis testing using surrogate data: Application to black hole light curves

“…7 taking time series from the Lorenz attractor (upper panel) and white noise (lower panel) and computing the degree distribution for various M values. It is also known [35] that the variation of the degree distribution with M for white noise is similar to that of the surrogate data. However, the above observations regarding convergence are subjective since the distributions for two M values can never coincide exactly.…”

“…This is achieved by the proper choice of the recurrence threshold ǫ and the embedding dimension M. We have recently shown [35] that the value of ǫ is closely related to the choice of M. We have also proposed a general method for the construction of RN from a time series, which we follow here. The basic criterion that we use to select the recurrence threshold ǫ is that the resulting RN should have a giant component and the ǫ value obtained using this criterion is approximately the same for different time series for a given M, due to the uniform deviate transformation.…”

“…3, we show the result of RN analysis of the Lorenz attractor time series and its surrogates using both CPL and CC as quantifying measures. For each M, we use the value of ǫ that satisfies the primary criterion of the existence of a giant component in the RN for computing the network measures, as discussed in detail in our scheme [35]. It is clear that the null hypothesis that the data comes from a linear stochastic process can be rejected with high confidence level in both cases.…”