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Complex Network Analysis of Recurrences

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Recurrence Quantification Analysis

Part of the book series: Understanding Complex Systems ((UCS))

Abstract

We present a complex network-based approach to characterizing the geometric properties of chaos by exploiting the pattern of recurrences in phase space. For this purpose, we utilize the basic definition of a recurrence as the mutual proximity of two state vectors in phase space (disregarding time information) and re-interpret the recurrence plot as a graphical representation of the adjacency matrix of a random geometric graph governed by the system’s invariant density. The resulting recurrence networks contain exclusively geometric information about the system under study, which can be exploited for inferring quantitative information on the geometric properties of the system’s attractor without explicitly studying scaling characteristics as in the case of “classical” fractal dimension estimates.

Similar as the established recurrence quantification analysis, recurrence networks can be utilized for studying dynamical transitions in non-stationary systems, as well as for automatically discriminating between chaos and order without the necessity of extensive computations typically necessary when inferring this distinction based on the systems’ maximum Lyapunov exponents. Moreover, we provide a thorough re-interpretation of two bi- and multivariate generalizations of recurrence plots in terms of complex networks, which allow tracing geometric signatures of asymmetric coupling and complex synchronization processes between two or more chaotic oscillators.

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Notes

  1. 1.

    Here, phase-coherent describes chaotic oscillations around a well-defined center in a suitably chosen two-dimensional projection of the system [111, 112], whereas such a projection is not possible in the funnel regime.

  2. 2.

    Here, \(\varepsilon\)-balls refers to general (hyper-)volumes according to the specific norm chosen for measuring distances in phase space, e.g., hypercubes of edge length \(2\varepsilon\) in case of the maximum norm, or hyperballs of radius \(\varepsilon\) for the Euclidean norm.

  3. 3.

    In Eq. (4.7) as well as several following definitions, we skip the condition i ≠ j ≠ v for simplicity, since A ii  = 0 for all vertices i according to our RN definition Eq. (4.4).

  4. 4.

    Strictly speaking, this is only true if the recurrence rate is defined such that the main diagonal in the RP is excluded in the same way as potential self-loops from the RN’s adjacency matrix.

  5. 5.

    It is important to realize that cross-recurrences are not to be understood in the classical sense of Poincaré’s considerations, since they do not indicate the return of an isolated dynamical system to some previously assumed state. In contrast, they imply an arbitrarily delayed close encounter of the trajectories of two distinct systems and, therefore, should be better named cross encounters instead. Following the same reasoning, terms such as cross-recurrence plot or cross-recurrence rate are suggestive, but potentially misleading. However, to comply with the existing literature on cross-recurrence plots, we will adopt the established terms even despite their conceptual ambiguities.

  6. 6.

    One corresponding strategy could be utilizing methods for community detection in networks [36], such as consideration of modularity [71]. Notably, such idea has not yet been explored in the context of RN analysis, and it is unclear to what extent the inferred possible community structure of an IRN could exhibit relevant information for studying any geometric signatures associated with the mutual interdependences between different dynamical systems. To this end, we leave this problem for future research.

  7. 7.

    In the specific case of an IRN, we interpret this as geometric signatures of the coupling between the underlying dynamical systems X k and X l [33, 34].

  8. 8.

    For ergodic systems, sampling from one long trajectory, ensembles of short independent realizations of the same system, or directly from the invariant density should lead to networks with the same properties at sufficiently large N.

  9. 9.

    In fact, we should take here the transitivity dimension of the RN obtained for X ⊗ Y, i.e., \(\hat{D}_{\mathcal{T}^{X\otimes Y }} =\log (\hat{\mathcal{T}}^{X\otimes Y })/\log (3/4)\), which is in general not identical to the pseudo-dimension \(\hat{\tilde{D}}_{\mathcal{T}^{J}}\) due to the different metrics used for the definition of recurrences of X ⊗ Y and joint recurrences of X and Y.

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Acknowledgements

The reported development of the recurrence network framework has been a community effort. Among other colleagues, we particularly acknowledge important contributions by Jobst Heitzig and Norbert Marwan, as well as multiple inspiring discussions with Jürgen Kurths. Financial support of this work has been granted by the German Federal Environmental Agency, the European Union Seventh Framework Program, the Max Planck Society, the Stordalen Foundation, the German Research Association via the IRTG 1740 “Dynamical phenomena in complex networks”, the National Natural Science Foundation of China (Grant No. 11305062, 11135001), and the German Federal Ministry for Science and Education (project CoSy-CC2, grant no. 01LN1306A). Numerical codes used for estimating recurrence network properties can be found in the software package pyunicorn [20], which is available at http://tocsy.pik-potsdam.de/pyunicorn.php.

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Authors and Affiliations

  1. Potsdam Institute for Climate Impact Research, Telegrafenberg A31, 14473, Potsdam, Germany

    Reik V. Donner, Jonathan F. Donges, Yong Zou & Jan H. Feldhoff

  2. Max Planck Institute for Biogeochemistry, Hans-Knöll-Straße 10, 07745, Jena, Germany

    Reik V. Donner

  3. Stockholm Resilience Centre, Stockholm University, Kräftriket 2B, 11419, Stockholm, Sweden

    Jonathan F. Donges

  4. Department of Physics, East China Normal University, 200062, Shanghai, China

    Yong Zou

  5. Department of Physics, Humboldt University, Newtonstraße 15, 12489, Berlin, Germany

    Jan H. Feldhoff

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  1. Reik V. Donner
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  4. Jan H. Feldhoff
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Correspondence to Reik V. Donner .

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Editors and Affiliations

  1. Department of Cell and Molecular Physiology, Loyola University, Maywood, Illinois, USA

    Charles L. Webber, Jr.

  2. Transdisciplinary Concepts and Methods, Potsdam Institute for Climate Research Research Domain, Potsdam, Germany

    Norbert Marwan

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Donner, R.V., Donges, J.F., Zou, Y., Feldhoff, J.H. (2015). Complex Network Analysis of Recurrences. In: Webber, Jr., C., Marwan, N. (eds) Recurrence Quantification Analysis. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-07155-8_4

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