Complex network approaches to nonlinear time series analysis

Jonathan F. Donges; J\"urgen Kurths; Norbert Marwan; Reik V. Donner; Yong Zou

REVIEW 2 major objections 3 minor 298 references

Complex network approaches to nonlinear time series analysis

T0 review · 2 major / 3 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Complex network representations of time series can estimate dynamical invariants—fractal dimension and Lyapunov-like exponents—more dependably than classical nonlinear methods, this review argues.

desk verdict A competent, self-aware reprint of the 2019 Physics Reports review; the flagship dimension-estimation claim is less proven than the intro suggests, but the paper itself flags the gap. read the letter →

arxiv 2501.18737 v1 pith:AHC6CCXK submitted 2025-01-30 physics.data-an math-phmath.MPnlin.CD

Yong Zou , Reik V. Donner , Norbert Marwan , Jonathan F. Donges , Jürgen Kurths This is my paper · ORCID

classification physics.data-anmath-phmath.MPnlin.CD

keywords complexnetworksnonlineartimeseriesanalysisrecurrencevisibilitygraphstransitionfractaldimensionLyapunovexponentrandomgeometric

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This review argues that turning a time series into a graph is not just a visualization trick: it is a genuine extension of nonlinear time series analysis. The authors organize the field into three families—recurrence networks built from phase-space proximity, visibility graphs built from local convexity, and transition networks built from symbolic dynamics—and show that each family carries information that classical methods struggle to extract. Their central examples are that the transitivity dimension of a recurrence network approximates the attractor's fractal dimension, and that the mean out-degree or diameter of an ordinal-pattern transition network performs similarly to the Lyapunov exponent. A sympathetic reader should care because these network quantities are computed from a single finite time series without requiring the delicate scaling-region fits that classical invariant estimates need.

What carries the argument

The load-bearing objects are the three time-series-to-network transformations. Recurrence networks take the recurrence matrix $R_{ij}(\varepsilon)=\Theta(\varepsilon-\|\vec{x}_i-\vec{x}_j\|)$ as an adjacency matrix, minus the diagonal; the equivalence to random geometric graphs then supplies analytic formulas for degree distributions, transitivity, and percolation, and the limit $\varepsilon\,l_{ij}(\varepsilon)\to g(\vec{x}_i,\vec{x}_j)$ links shortest paths to geodesics on the attractor. The transitivity dimension $\mathcal{D}_T=\lim \log T(\varepsilon)/\log(3/4)$, with upper and lower variants, inverts the known dependence of random-geometric-graph transitivity on dimension. Visibility graphs connect observations that see each other, encoding local convexity and record statistics, with directed variants that test time-series irreversibility. Transition networks coarse-grain phase space or ordinal patterns and count transition frequencies, giving Markov-chain-like graphs whose out-degrees and diameters act as dynamical indices.

What would settle it

On a two-dimensional chaotic map with a known fractal dimension, estimate the transitivity dimension from a single long trajectory and from an ensemble of many short independent trajectories. If the estimates differ beyond finite-sample scatter, or if neither matches the known dimension, the ergodic-sampling premise and the claimed geometric interpretation are not supported.

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Extended reading notes

Core claim

The paper's central claim is that complex network approaches can partially solve fundamental, long-standing problems that other time series methods have not successfully addressed. Concretely, recurrence networks—graphs whose edges connect state vectors closer than a threshold $\varepsilon$—are random geometric graphs embedded in the attractor, so their transitivity and local clustering define upper and lower transitivity dimensions and clustering dimensions that approximate the attractor's fractal dimension. Ordinal-pattern transition networks, whose vertices are rank-order patterns and whose edges are observed successions, yield mean out-degrees and diameters that track the Lyapunov exponent. The review further claims that these three network families provide complementary geometric, ordering, and symbolic information, and it assembles practical guidance on embedding, thresholds, noise, and non-stationarity that makes the methods usable beyond toy models.

Load-bearing premise

The load-bearing premise is that one sufficiently long trajectory samples the attractor's invariant density representatively, and that a valid time-delay embedding exists; if the dynamics are not ergodic or the embedding is invalid, the network measures lose their claimed geometric and dynamical meaning.

Editorial extensions

If this is right

If the central claim is right, fractal dimension and Lyapunov-like stability can be estimated from recurrence and ordinal-pattern networks without selecting a linear scaling region, removing a notoriously subjective step in nonlinear time series analysis.
Network measures such as transitivity and average path length can serve as normalized, system-independent discriminators between periodic and chaotic regimes, including in parameter sweeps and sliding-window analyses of slowly drifting systems.
The same toolbox transfers to multivariate settings: multiplex recurrence networks and joint or cross recurrence networks detect synchronization and coupling geometry in coupled systems.
Visibility-graph degree statistics and time-directed decompositions provide practical tests for long-range correlations and time-series irreversibility on unevenly sampled or noisy data.
Because the methods are collected in a modular software package, the review's recommendations translate directly into applied analysis across climatology, physiology, engineering, and economics.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: if transitivity dimensions are as stable as claimed, sliding-window recurrence networks could provide online bifurcation and tipping-point detectors that flag regime shifts without pre-specifying embedding parameters—a use the review notes but does not develop.
Beyond the paper: the random-geometric-graph analogy suggests a surrogate test—compare the observed transitivity dimension against the analytic value for the best-fitting invariant density—which would turn network measures into goodness-of-fit statistics for hypothesized dynamics.
Beyond the paper: combining visibility-graph scaling with multiplex layers across time scales could yield a multifractal characterization of a single series, extending the review's univariate long-range-correlation estimates.
Testable extension: on noisy chaotic data, direct comparison of recurrence-network dimension estimates with the classical correlation-dimension algorithm as a function of noise level and series length would make the claimed stability advantage precise.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This is a straight reprint of the 2019 Physics Reports review, and it is a good review on its own terms. The taxonomy of time series networks — recurrence networks, visibility graphs, transition networks — is thorough, the random-geometric-graph framing is clearly explained, and the practical guidance on thresholds, embedding, noise, and non-stationarity is genuinely useful. The survey of applications and the pyunicorn software section also add real value. Credit where due: the paper is honest about most of its limitations, and the analytical core for smooth manifolds is coherent. The stress-test concern about the transitivity dimension is legitimate but not fatal. Section 3.4 explicitly restricts the continuum limit to compact smooth manifolds and says fractal sets S will not be considered. Yet Section 3.5.5 applies the transitivity dimension to the Hénon map and Rössler system — fractal attractors — and the paper concedes that a detailed analytical investigation of continuous versus fragmented invariant densities is future work. So the headline claim in Section 1.1 that recurrence-network transitivity gives a more robust estimation of fractal dimension rests on numerical evidence rather than on the RGG theory advertised as its foundation. The review does not hide this, but the intro states it more strongly than the later caveats justify. A careful reader should treat the transitivity dimension as a heuristic complexity indicator, not a proven dimension estimator. Minor soft spots: the self-citation density is high, though the cited results are real and independently anchored; and the phrase 'more robust estimation of dynamical invariants' overpromises relative to what is actually demonstrated. Neither undermines the review's usefulness. For whom: anyone entering this subfield or wanting a single reference on time series network methods. It is a review, so there is no new result; if the venue wants original research, desk reject on novelty. But as a review it deserves serious refereeing, and I would cite it. I would not bring the whole 97 pages to a reading group, but a chapter or two would be fine.

Referee Report

2 major / 3 minor

Summary. This manuscript is a comprehensive review of complex-network representations of time series, organized around three main families: phase-space recurrence networks, visibility graphs, and Markov/ordinal-pattern transition networks. It covers the methodological definitions, network-theoretic measures, analytical connections to random geometric graphs, practical parameter-selection issues (threshold, embedding, noise, non-stationarity), multivariate extensions, real-world applications, and a software overview. The central claim is that these network approaches are not merely alternative visualizations but can supplement and partially solve long-standing problems in nonlinear time series analysis, with two headline examples: estimating fractal dimension from recurrence-network transitivity/clustering dimensions, and estimating Lyapunov-type instability from ordinal-pattern transition network measures.

Significance. If its central claims are accepted, the review is a valuable synthesis of a large and active literature, and the random-geometric-graph framework genuinely clarifies the geometric content of recurrence-network measures. The paper is also unusually transparent about practical limitations: it discusses threshold selection, embedding dependence, noise robustness, non-stationarity, and explicitly identifies open theoretical questions. Its strengths include the systematic taxonomy of network types, the careful distinction between geometric and dynamical information, and the concrete guidance on algorithmic choices. The main weakness is that one of the two headline claims in Section 1.1 — robust estimation of fractal dimension via transitivity dimensions — is stated more strongly than the analytical foundation presented in Sections 3.4 and 3.5.5 supports.

major comments (2)

[§1.1, §3.4.2, §3.5.5 (Eqs. 50–53)] The headline claim that recurrence-network transitivity and local clustering dimensions provide a more robust estimation of fractal dimension is not supported by the analytical framework that the review itself presents. Section 3.4.2 explicitly restricts the continuum-limit framework to compact smooth manifolds and states that the limit 'may not be assessible in the case of fractal sets S, which we will not further consider.' The classical random-geometric-graph transitivity result cited from [150] is for integer-dimensional metric spaces. Nevertheless, Eqs. (50)–(53) are applied to the Hénon and Rössler attractors, with the text immediately conceding that a detailed analytical investigation of the differing behavior for continuous versus fragmented invariant densities 'will be subject of future work.' Because this is one of only two concrete examples in Section 1.1 of the review's central assertion, the manuscript should either explicitly downgrade the claim to a numerically validated heuristic for fractal attractors, or supply the missing analytical argument linking these limits to the fractal dimension. Without one of these changes, the 'more robust estimation' statement overstates the current theoretical basis.
[§3.5.5 (Eqs. 50–53) and Fig. 9] The text says that for systems without fragmented invariant density the upper and lower transitivity dimensions 'practically coincide,' allowing estimation from a single network instance at one suitably chosen ε. For the Hénon map, which has a fragmented invariant density, Fig. 9(a) shows ε-dependent oscillations between two accumulation points, so a single-ε estimate does not define either D_T^u or D_T^l uniquely. Since the Hénon map is one of the paper's flagship examples for the dimension-estimation claim, the review should explicitly warn that for such attractors the transitivity dimension is defined only through the two limits over ε and that a single-network, single-threshold estimate can be ambiguous.

minor comments (3)

[§3.3, first paragraph] The statement that RN analysis can be performed using only a single fixed scale ε 'instead of explicitly studying scaling properties over a range of threshold values' is in tension with Section 3.5.5, where the transitivity and clustering dimensions are defined as limits over ε. This sentence should be qualified to say that most RN measures are computed at a single scale, whereas the dimension estimators require systematic scale variation.
[Nomenclature and §3.5.5] The notation for the dimension estimators is not fully consistent: the table of abbreviations lists \hat{D}_C and \hat{D}_T, while Eqs. (50)–(53) use D_T^u, D_T^l, D_C^u, D_C^l without the hat. Using one consistent notation would improve readability.
[General formatting] The text contains numerous typographical and OCR artifacts (for example, 'dara minimg' in the introduction and garbled symbols in several display equations). A careful copyediting pass is needed before publication.

Circularity Check

2 steps flagged · score 4.0 of 10

The flagship fractal-dimension claim rests on a (3/4)^D ansatz that the paper's own analytic section excludes and defers; the fractal-case bridge is a self-citation [40].

ansatz smuggled in via citation [Section 3.5.5 (Eqs. (50)–(53)) vs. Section 3.4.2 and Section 1.1]
"Continuous analogs ... should be approximated by taking the limit N → ∞ and ε → 0 (note that the latter limit may not be assessible in the case of fractal sets S, which we will not further consider in the following). ... This analytical relationship can be generalized to attractor manifolds with non-integer fractal dimensions, which can in turn be estimated from the RN transitivity by inverting this function. ..."

The flagship claim (§1.1) makes RN transitivity/clustering dimensions estimate fractal dimension. The only proven link in the review is the integer-dimensional RGG law T=(3/4)^m [150]. The load-bearing extension T≈(3/4)^D to non-integer fractal dimension is asserted in one sentence ('can be generalized to attractor manifolds with non-integer fractal dimensions') and attributed to [40], the authors' own prior work — while §3.4.2 says the continuum limit 'may not be assessible in the case of fractal sets S, which we will not further consider,' and §3.5.5 calls the analytic treatment of fragmented invariant densities 'subject of future work.' Because Eqs.
self definitional [Section 3.5.5, paragraph following Eqs. (50)–(53)]
"Notably, the analytical relationship in Eqs. (50), (51) between the effective (geometric) dimension of chaotic attractors and RN transitivity provides the theoretical justification and foundation for applying T as a characteristic discriminating between high and low dynamical complexity of chaotic attractors."

Eqs. (50)–(53) are the definitions of the upper/lower transitivity dimensions (log T / log(3/4)); the paper itself introduces them as the 'two definitions' of these quantities. Invoking these definitions as 'the analytical relationship ... between the effective (geometric) dimension of chaotic attractors and RN transitivity' converts the justification for using T as a complexity discriminator into the definition plus the assumed (3/4)^D scaling. By construction, D_T equals the dimension m whenever the RGG law holds, so the claimed 'theoretical justification' is the calibration of the estimator on integer-dimensional RGGs, with the fractal case still unproven by the paper's own account.

full rationale

This review is largely a survey, and most of its content is a faithful synthesis of results derived elsewhere. I found no instance of a fitted parameter being renamed as a prediction, and I did not flag the visibility-graph Hurst or ordinal-pattern-transition-network Lyapunov claims because the provided text does not exhibit their derivations. The one genuinely load-bearing circular element concerns the flagship claim of Section 1.1: RN transitivity and local clustering dimensions are said to approximate the fractal dimension of chaotic attractors and thereby 'solve partially some fundamental and long standing problems' of robustly estimating dynamical invariants. Within the paper's own derivation chain, the estimator D_T = log T / log(3/4) (Eqs. 50–53) is the inversion of the RGG law T = (3/4)^m, which is proven only for integer-dimensional, uniform, max-norm settings (Dall–Christensen [150]). Section 3.4.2 explicitly states that the continuum limit 'may not be assessible in the case of fractal sets S,' and Section 3.5.5 concedes that the ε-dependence of transitivity for attractors with fragmented invariant densities 'will be subject of future work.' The generalization to non-integer fractal dimension is asserted in a single sentence and attributed to [40], the authors' own prior work; the in-review figure (Fig. 9) validates only that finite-N estimates converge to the estimator's own limsup/liminf, not agreement with independently known dimension values. In this sense the central 'prediction' of fractal dimension is, at the level of the review's own derivation, the assumed scaling law restated by definition, with the gap filled by self-citation. This is partial circularity, not wholesale: the integer-dimension RGG theory is a genuine external theorem, and [40]'s numerical comparison against the independently known correlation dimension of the Hénon and Rössler attractors is externally falsifiable evidence, so the central claim retains real independent content. The paper is also candid about several related gaps (degree-distribution power laws are not generally related to fractal dimension; the average path length has reduced explanatory power), which weighs against a higher score. Overall, the loading-bearing self-citation and definitional construction justify a moderate circularity score, but the independent numerical content keeps it well below the 'forced by definition' range.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The review introduces no new fitted parameters or invented physical entities. The only free choices are algorithmic parameters such as the recurrence threshold, embedding dimension, and delay, which the review itself identifies as user-selected and discusses at length. The central claims rest on standard embedding theory, ergodicity assumptions, and the random geometric graph analogy, all explicitly stated in the text.

assumptions (4)

standard math Takens' time-delay embedding theorem: for deterministic dynamics, an m-dimensional embedding is topologically equivalent to the original phase space if m is sufficiently large relative to the attractor dimension.
Invoked in Section 3.1.1 to justify reconstructing phase space from a scalar time series before building recurrence networks.
domain assumption Sampled state vectors can be treated as draws from the invariant density of the attractor, relying on ergodicity and suitable sampling conditions.
Used in Sections 3.4.1 and 3.5.1 to identify recurrence networks with random geometric graphs and to derive the degree distribution formula.
domain assumption Recurrence networks are random geometric graphs on the attractor manifold, with adjacency defined by a hard distance threshold.
This equivalence is the theoretical backbone of Section 3.4 and underlies the interpretation of transitivity and clustering as dimension estimators.
domain assumption The transitivity of a random geometric graph decays as (3/4)^D with geometric dimension D, allowing dimension estimation by inverting this relationship.
Adopted from the cited literature in Section 3.5.5 and used for the transitivity and clustering dimensions, Eqs. (50)-(53).

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Cite this review

Pith. "Pith review of Complex network approaches to nonlinear time series analysis." pith.science (2026). https://pith.science/paper/AHC6CCXK

@misc{pith2026250118737,
  author       = {Pith},
  title        = {Pith review of: Complex network approaches to nonlinear time series analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AHC6CCXK}},
  note         = {Machine review of arXiv:2501.18737}
}

read the original abstract

In the last decade, there has been a growing body of literature addressing the utilization of complex network methods for the characterization of dynamical systems based on time series. While both nonlinear time series analysis and complex network theory are widely considered to be established fields of complex systems sciences with strong links to nonlinear dynamics and statistical physics, the thorough combination of both approaches has become an active field of nonlinear time series analysis, which has allowed addressing fundamental questions regarding the structural organization of nonlinear dynamics as well as the successful treatment of a variety of applications from a broad range of disciplines. In this report, we provide an in-depth review of existing approaches of time series networks, covering their methodological foundations, interpretation and practical considerations with an emphasis on recent developments. After a brief outline of the state-of-the-art of nonlinear time series analysis and the theory of complex networks, we focus on three main network approaches, namely, phase space based recurrence networks, visibility graphs and Markov chain based transition networks, all of which have made their way from abstract concepts to widely used methodologies. These three concepts, as well as several variants thereof will be discussed in great detail regarding their specific properties, potentials and limitations. More importantly, we emphasize which fundamental new insights complex network approaches bring into the field of nonlinear time series analysis. In addition, we summarize examples from the wide range of recent applications of these methods, covering rather diverse fields like climatology, fluid dynamics, neurophysiology, engineering and economics, and demonstrating the great potentials of time series networks for tackling real-world contemporary scientific problems.

Figures

Figures reproduced from arXiv: 2501.18737 by the authors.

**Figure 2.** Graphical representation of the different complex [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Schematic illustration of some characteristics o [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

Figures from the paper (33 more)

**Figure 4.** Figure 4: Basic concepts beyond recurrence plots and the res [PITH_FULL_IMAGE:figures/full_fig_p022_4.png]

**Figure 5.** Figure 5: A graphical representation of the Lorenz attracto [PITH_FULL_IMAGE:figures/full_fig_p023_5.png]

**Figure 6.** Figure 6: Spatial distributions of vertex characteristics [PITH_FULL_IMAGE:figures/full_fig_p027_6.png]

**Figure 7.** Figure 7: Schematic illustration of a set S (gray), where g(~x, ~y) denotes the geodesic distance between ~x, ~y ∈ S (after [145]). (see Section 3.5.5). In the following, we review a corresponding analytical framework for general spatially embedded networks which is specifically…

**Figure 8.** Figure 8: (a) Complementary cumulative distribution funct [PITH_FULL_IMAGE:figures/full_fig_p034_8.png]

**Figure 9.** Figure 9: (a) Transitivity dimensions Dˆ T of the H´enon map (Eq. A.4 for one realization with initial condition (x, y) = (0, 0), the first 1000 iterations have been removed from the trajectory to avoid transient behavior) for different N. Dashed horizontal lines indicate numeri…

**Figure 10.** Figure 10: Dependence of the average path length L (a-c) and the global clustering coefficient C (d-f) for the H´enon map (Eqs. A.4), R¨ossler (Eqs. A.2), and Lorenz system (Eqs. A.1) (from left to right). Dashed lines in the plots on Lˆ(ε) indicate the approximate presence of t…

**Figure 11.** Figure 11: RN transitivity Tˆ (A) and average path length Lˆ (B) for a two-dimensional intersection (a = b) of the threedimensional parameter space of the R¨ossler system (Eq. A.2), displaying “shrimp” structures (i.e., self-similar periodic windows with complex shape). For det…

**Figure 12.** Figure 12: Phase space of the standard map (Eq. A.7) characterized by three RN measures for the standard map using fixed RR = 0.02. (a) Tˆ , (b) Cˆ, (c) Lˆ. Reproduced from [176]. we choose 200 initial conditions distributed randomly within the domain of definition of the standa…

**Figure 13.** Figure 13: Dependence of (a) RN transitivity Tˆ and (b) global clustering coefficient Cˆ for fGn on the Hurst exponent H for different embedding dimensions (m = 3: , m = 4: ⊳, m = 5: ∗, m = 6: •), taken over 200 independent realizations and using a RN edge density of ρ = 0.03. T…

**Figure 14.** Figure 14: Construction of multiplex recurrence networks for multivariate time series. Reproduced from [100] with permission. and squeezed along certain directions, so that the resulting geometric structure appears significantly lowerdimensional than m. More numerical considera…

**Figure 15.** Figure 15: Schematic representation of cross-recurrence ( [PITH_FULL_IMAGE:figures/full_fig_p049_15.png]

**Figure 16.** Figure 16: Global cross-clustering coefficients (Eq. 22) [PITH_FULL_IMAGE:figures/full_fig_p054_16.png]

**Figure 17.** Figure 17: Joint transitivity Tˆ J , single-system RN transitivities Tˆ X,Y (Eq. 11), corresponding transitivity dimensions Dˆ T X , Dˆ T Y (Eq. 50) and derived dimensional locking index DLI [] (Eq. (83)) (from top to bottom) for two unidirectionally coupled R¨ossler systems (X …

**Figure 18.** Figure 18: Schematic illustration of the algorithm for cons [PITH_FULL_IMAGE:figures/full_fig_p063_18.png]

**Figure 19.** Figure 19: (color online) (a, b) Estimates of λ for approximately exponential degree distributions of the AR(1) process. (c, d) λ versus ϕ1. (a, c) VG, and (b, d) HVG. Each dot in panels (c) and (d) represents an average over 50 independent random realizations of 5000 data point…

**Figure 20.** Figure 20: The Kolmogorov-Smirnov (KS) test statistics [PITH_FULL_IMAGE:figures/full_fig_p071_20.png]

**Figure 21.** Figure 21: As in Fig. 20 for quantifying the effect of additive [PITH_FULL_IMAGE:figures/full_fig_p071_21.png]

**Figure 22.** Figure 22: As in Fig. 21 for fixed data, but uncertain timing of [PITH_FULL_IMAGE:figures/full_fig_p072_22.png]

**Figure 23.** Figure 23: Time irreversibility testing for white noise (a, [PITH_FULL_IMAGE:figures/full_fig_p077_23.png]

**Figure 24.** Figure 24: (color online) Frequency distributions of [PITH_FULL_IMAGE:figures/full_fig_p079_24.png]

**Figure 25.** Figure 25: (a) Illustration of permutation symbols from a ti [PITH_FULL_IMAGE:figures/full_fig_p084_25.png]

**Figure 26.** Figure 26: (a) R¨ossler attractor in phase space color coded [PITH_FULL_IMAGE:figures/full_fig_p086_26.png]

**Figure 27.** Figure 27: Phase synchronization transitions of three coup [PITH_FULL_IMAGE:figures/full_fig_p088_27.png]

**Figure 28.** Figure 28: Ordinal transition networks on the path to phase s [PITH_FULL_IMAGE:figures/full_fig_p088_28.png]

**Figure 29.** Figure 29: Cross and joint OPTNs, which are reconstructed from two coupled R¨ossler systems (Eqs. (A.5)) in the non-synchronized regime [356]. (a) Normal cross ordinal pattern transition network (COPTN), (b) alternative version of the COPTN, and (c) joint ordinal pattern transit…

**Figure 30.** Figure 30: (a) Terrestrial dust flux records from the three co [PITH_FULL_IMAGE:figures/full_fig_p094_30.png]

**Figure 31.** Figure 31: Results of sliding window HVG-based tests for tim [PITH_FULL_IMAGE:figures/full_fig_p097_31.png]

**Figure 32.** Figure 32: p(k) of VGs from monthly (a,c) and daily data (b,d). (a,b) is for {xi}, and (c,d) {−xi}. One would suspect a fit to the first part of p(k−x) yields that the slope of dashed line in (c) is 1.79, and that of (d) is 3.61, but all p-values are 0, rejecting the hypothetica…

**Figure 33.** Figure 33: Network representations of the VG constructed fr [PITH_FULL_IMAGE:figures/full_fig_p099_33.png]

**Figure 34.** Figure 34: (a) Absolute excess degrees ∆ki obtained from the VGs of A N,S computed over the sliding windows with a width of w = 270 months and a mutual overlap of 12 months. Error bars display mean values and standard deviations within a given time window centered at the respect…

**Figure 35.** Figure 35: Bifurcation diagram of the diode resonator data w [PITH_FULL_IMAGE:figures/full_fig_p102_35.png]

**Figure 36.** Figure 36: (a) Scatter plot of global node out-link entropy [PITH_FULL_IMAGE:figures/full_fig_p104_36.png]

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