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Complex network approaches to nonlinear time series analysis

Yong Zou, Reik V. Donner, Norbert Marwan, Jonathan F. Donges, Jürgen Kurths

arXiv:2501.18737v1physics.data-anmath-phnlin.CD

TL;DR

The report addresses the limited overlap between time-series mining and nonlinear time-series analysis using complex networks. It reviews recurrence networks, visibility graphs, and transition networks, showing their theoretical insights, applications, and practical limitations.

  • Problem

    Time-series mining has had practically no overlap with nonlinear time-series analysis using complex network methods, despite the practical relevance of both areas.

  • Method

    The report reviews network representations of time series based on recurrences, visibility graphs, and Markov chains, including their variants, interpretations, and practical considerations.

  • Results

    Complex network approaches provide valuable insights from dynamical-systems and stochastic-process perspectives and support applications across diverse fields.

  • Takeaways & Limitations

    Time-series networks offer a broad framework for characterizing dynamical systems and addressing contemporary scientific problems across climatology, fluid dynamics, neurophysiology, engineering, and economics.

  • Takeaways & Limitations

    Coarse-graining transition networks loses detailed temporal information and requires stationarity and ergodicity for a static representation.

Abstract

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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.

1. Introduction

The introduction positions complex-network methods as a dynamical-systems-based complement to nonlinear time-series analysis, addressing gaps left by conventional data-mining and nonlinear techniques. The review surveys network representations that yield additional dynamical insights and applications across diverse fields.

  • Introduction: Time-series analysis differs from general data analysis because observations possess a natural temporal ordering.This ordering distinguishes time-series methods from cross-sectional and spatial analyses.
  • Complex network approaches: The review addresses limited overlap between data-mining tools and dynamical-systems-based complex-network approaches for nonlinear time-series analysis.It treats network methods as applications of complex-network theory to synthetic and experimental series.
  • Nonlinear time series analysis: Nonlinear time-series analysis characterizes complex temporal evolution using concepts and techniques originating in dynamical-systems and chaos theory.Its framework includes geometric and dynamical descriptions of invariant measures in phase space.
  • Nonlinear time series analysis: Established nonlinear methods face practical challenges involving dimensionality, non-stationarity, embedding choices, finite data, noise, irregular sampling, and computational complexity.Parameter selection and the visibility of scaling regimes can substantially affect estimated dynamical invariants.
  • Complex network approaches: The reviewed network approaches can provide complementary dynamical features and more robust estimates of invariants such as fractal dimensions and Lyapunov exponents.Examples include transitivity and local clustering dimensions from recurrence networks and ordinal-transition measures related to the Lyapunov exponent.

2. Complex network theory

This section introduces graphs as representations of vertices and edges, then develops standard measures for local, mesoscopic, and global connectivity. It also covers paths, edge structure, motifs, and the small-world property.

  • Preliminaries: A complex network is represented as a graph G = (V, E), with vertices representing elements and edges representing connections between vertex pairs.Graphs may be directed or undirected, and edges may carry weights.
  • Preliminaries: Thresholding a weighted matrix W produces a binary adjacency matrix A whose nonzero entries identify graph edges.This construction can additionally impose symmetry on A.
  • Path-based measures: Path-based measures such as closeness, local efficiency, and betweenness characterize connectivity and vertex importance across the network.Betweenness measures the fraction of shortest paths traversing a vertex, including applications to information propagation.
  • Network measures: Degree, clustering, and degree-distribution measures characterize local and mesoscopic connectivity, while network entropy summarizes degree-distribution heterogeneity.The local clustering coefficient measures mutual connections among a vertex’s neighbors.
  • Edge and meso-scale structure: Edge-oriented measures include matching index, edge betweenness, and network motifs, which capture neighborhood overlap, shortest-path participation, and meso-scale connection patterns.Motifs generalize local clustering and help classify networks by recurring local structures.
  • Stylized facts of complex networks: Small-world networks combine short average path lengths with high clustering relative to random graphs.In regular d-dimensional hypercubic lattices, mean path length grows as N^(1/d).

3. Recurrence networks in phase space

Recurrence networks reinterpret phase-space recurrences as geometric network structure, extending recurrence plots with network representations and measures. Their construction depends on reconstructing an appropriate phase space and choosing a recurrence threshold.

  • Recurrence networks: Recurrence networks study recurrences in phase space geometrically, complementing recurrence quantification and other recurrence-based methods.The section emphasizes geometric characteristics associated with the structural organization of the underlying dynamical system.
  • Phase-space reconstruction: A scalar time series is commonly transformed into m-dimensional delay-embedding vectors using an embedding delay τ.The embedding vector is defined as x⃗_i = (x_i, x_i−τ, · · ·, x_i−(m−1)τ).
  • Phase-space reconstruction: Embedding dimension m and delay τ are not known a priori, so false nearest neighbors and dependence-based criteria provide practical selection approaches.The review cautions that embedding suitability and analysis dependence on embedding parameters should be checked for experimental data.
  • Recurrence plots: A recurrence plot marks pairs of phase-space states as recurrent when their distance is below threshold ε, using a chosen phase-space norm.The threshold determines whether two state vectors are considered close, while the recurrence matrix provides the two-dimensional representation.
  • Network variants: Recurrence-network studies have also examined variants such as k-nearest-neighbor representations and fuzzy recurrence plots.These alternatives modify geometric closeness or emphasize recurrence regions and structures in the phase-space representation.
  • Network construction: An ε-recurrence network treats sampled state vectors as vertices and connects pairs that are mutually close in phase space.The recurrence matrix is reinterpreted as a network adjacency matrix, excluding the main diagonal or self-loops.

3.3. Complex network characteristics of RN

Recurrence-network characteristics translate graph structure into local and global geometric information about the sampled attractor. Measures describe density, dimensionality, centrality, fragmentation, neighborhood overlap, and recurrence rate.

  • Geometric interpretation: Recurrence-network measures provide geometric rather than dynamical characteristics because network properties are generally invariant under vertex relabelling.This makes RN analysis complementary to recurrence methods that retain temporal ordering.
  • Local vertex measures: The local degree density estimates the density of sampled states within an ε-ball around a phase-space state.High degree density identifies phase-space regions with high sampled residence probability.
  • Local vertex measures: The local clustering coefficient measures mutual closeness among neighbors and is associated with local geometric alignment and effective dimensionality.Near low-period unstable periodic orbits, effectively lower-dimensional dynamics produce more triangles and higher local clustering.
  • Centrality measures: Closeness and local efficiency are highest for vertices located near the center of the recurrence network.These measures quantify geometric proximity to other states through network distances.
  • Centrality measures: Betweenness centrality indicates local attractor fragmentation, with sparsely populated regions forming geometric bottlenecks between dense regions.Edge betweenness likewise characterizes local fragmentation in phase space.
  • Pairwise measures: Matching index measures neighborhood overlap, whereas edge betweenness measures path-bundling, so no simple correspondence exists between them.For pairs where both are nonzero, the review reports a clear anti-correlation because close states can be exchanged on shortest paths.
  • Global measures: For recurrence networks, edge density equals the recurrence rate when both exclude the main diagonal consistently, and it increases monotonically with ε.Increasing ε adds neighbors and therefore increases the number of edges.

3.4. Analytical theory of RN

The analytical theory of recurrence networks exploits their equivalence to random geometric graphs to connect finite network measures with continuous geometric properties of attractors. In the continuum limit, shortest network paths approximate attractor geodesics.

  • Random geometric graph foundation: Recurrence networks are specific random geometric graphs whose vertices sample an attractor according to an invariant density and connect through distance thresholds.This equivalence enables analytical treatment of RN characteristics using graph theory and computational geometry.
  • Continuum formulation: RN graph measures can be interpreted as discrete approximations of continuous geometric properties defined on the attractor and its invariant density.The framework establishes continuous analogs for degree, clustering, centrality, and pairwise measures.
  • Assumptions: The continuum framework assumes a compact smooth attractor set with a non-vanishing continuous invariant density and excludes the fractal-set limit from further consideration.Continuous analogs are formed by taking N →∞ and ε →0 under these assumptions.
  • Shortest paths and geodesics: In the limits N →∞ and ε →0, recurrence-network shortest paths approximate global geodesics on the invariant set.Geodesic distance is defined as the shortest path length among curves on the attractor under the chosen metric.
  • Local measures: Continuous degree density gives the probability that a point sampled from invariant density p falls within an ε-neighborhood of a fixed attractor point.Its discrete estimator is the classical RN degree density.
  • Local measures: Continuous clustering measures the probability that two points sampled near a reference point are mutually ε-close, approximated discretely by local clustering.Continuous closeness and efficiency analogously quantify geometric proximity through expected inverse geodesic distances.
  • Centrality measures: Continuous betweenness measures the probability that a point lies on a randomly chosen global geodesic, with RN betweenness as its discrete estimator.The construction requires geodesic multiplicities, which may be nonunique and even uncountable depending on the attractor geometry.

3.5. General properties of recurrence networks

General recurrence-network properties connect network statistics to invariant-density structure, attractor geometry, and dynamical complexity. The review also identifies important boundaries: scale choice affects discrimination, and several broad network phenomena do not transfer directly to RNs.

  • Degree distributions: Analytical degree-distribution results require ergodicity, sampling close to the attractor, and a sampling interval co-prime to system periods.Under these conditions, RN vertices can be treated as randomly distributed according to the invariant density.
  • Degree distributions: Power-law RN degree distributions arise when the invariant density has a power-law singularity, but excessively large ε can mask this behavior.The result is established for sufficiently small thresholds in the large-sample limit, while general higher-dimensional conditions remain unresolved.
  • Degree distributions: RN power laws reflect the local shape of the invariant density rather than a global fractal structure of the dynamical system.The review notes that scaling exponents may nevertheless coincide with fractal dimensions in particular systems.
  • Small-world effect: RNs do not obey small-world effects because, for fixed ε, average path lengths remain bounded independently of network size rather than scaling as log N.The average path length instead scales as ε^-1, and tuning ε can produce different target path lengths.
  • Mixing properties: RN mixing is comparatively understudied, although RNs often show assortative mixing when the invariant density is continuous or differentiable.This tendency is supported by numerical results cited in the review.
  • Dynamical complexity: Global network characteristics may provide more stable and distinctive indicators of dynamical complexity than some statistics based on local degree distributions.For time-continuous systems, spatial filling of the populated phase-space volume helps explain the discriminatory behavior of average path length.
  • Dynamical complexity: RN transitivity has an analytical relationship with effective attractor dimension, supporting its use for distinguishing high and low dynamical complexity.Unlike average path length, transitivity behaves qualitatively similarly for discrete and continuous systems and is normalized.

3.6. Practical considerations

Recurrence-network analysis requires careful choices of threshold, embedding, and data treatment because these parameters shape connectivity, topology, and interpretability. Robustness must be assessed against nonstationarity, extremes, noise, and measurement uncertainty.

  • Choice of recurrence rate or threshold: A recurrence threshold that is too small creates sparse, disconnected networks, whereas a threshold that is too large obscures fine geometric structure.A common rule of thumb is an edge density ρ ≲0.05, but threshold choice remains system- and embedding-dependent.
  • Choice of recurrence rate or threshold: The unique turning point of recurrence density versus threshold is not generally a valid selection rule because many systems exhibit several turning points.Surrogate-assisted optimization of a quality-loss function may be preferable for some civil-engineering signals.
  • Choice of recurrence rate or threshold: Normalizing amplitudes or fixing recurrence density helps sliding-window analyses handle varying fluctuation amplitudes, but extreme observations can still disconnect recurrence networks.Disconnected nodes can make measures such as average path length infinite or require artificial conventions.
  • Dependence on embedding parameters: For nonstationary fractional Brownian motion, phase-space reconstruction and low-dimensional dynamics do not apply even approximately, limiting the physical interpretation of recurrence-network results.Transformations such as detrending, deseasonalization, or differencing are required to remove the relevant nonstationarity.
  • ε-dependence of RN properties: Average path length decreases approximately as 1/ε, while clustering has a more system-dependent threshold dependence and rises approximately linearly at intermediate ε.At very small ε, finite samples can produce disconnected clusters.
  • Stability and robustness against noise: Moderate white or colored noise can preserve attractor-shape information, but topology may change above 50% signal-to-noise contamination and discrimination can fail at lower noise levels.RN measures fail to distinguish noisy periodic from noisy chaotic dynamics above 40% noise for C and 20% for L.
  • Stability and robustness against noise: Experimental uncertainty requires probabilistic recurrence relations rather than the binary recurrence decisions used in traditional analysis.The framework incorporates measurement uncertainty and uncertainty in observation timing.

3.7. Numerical examples

Numerical examples show that recurrence-network measures can distinguish dynamical regimes and track changes in attractor geometry across deterministic, Hamiltonian, and stochastic systems. Their interpretation depends on trajectory type, threshold selection, and embedding choices.

  • Rössler system: Recurrence networks distinguish periodic and chaotic Rössler solutions in parameter space, with periodic windows showing higher transitivity and average path length.The parameter-space plots display self-similar “shrimp” structures associated with periodic windows.
  • Rössler system: Recurrence-network properties also capture changes in Rössler attractor shape and invariant density across the transition from phase-coherent to funnel dynamics.Phase coherence can be characterized geometrically, although no theoretically derived RN-based phase-coherence index exists.
  • Standard map: For the standard map, recurrence networks distinguish coexisting regular and chaotic orbits and reliably detect the geometric organization of sticky orbits from relatively short time series.The analysis uses 200 initial conditions, 5000 time steps, and thresholds adapted to a fixed recurrence rate.
  • Standard map: Average path length in standard-map recurrence networks is strongly affected by orbit size and by whether a common threshold or fixed recurrence rate is used.Fixing recurrence rate compensates for differences in spatial distances when comparing trajectories.
  • Stochastic systems: Non-stationary systems require special care because standard phase-space reconstruction and low-dimensional dynamical assumptions do not apply directly to fractional Brownian motion.Differencing can transform fractional Brownian motion into stationary fractional Gaussian noise while retaining long-range correlations.
  • Stochastic systems: For fractional Gaussian noise with H > 0.5, transitivity and clustering depend little on the Hurst exponent at fixed embedding dimension, whereas for H < 0.5 both increase as H decreases.The low-H behavior reflects the non-optimal delay τ = 1 and stronger lag-one anticorrelation in anti-persistent processes.

3.8. Multiplex recurrence networks

Multiplex recurrence networks represent each component of a multivariate time series as a recurrence-network layer sharing time-indexed vertices. Layer similarity and coherence can then be quantified from their topology and projected into weighted networks.

  • Construction: A multiplex recurrence network contains one recurrence-network layer per time-series component, with identical time-indexed vertices and interlayer links only between counterpart nodes.The full multilayer structure is represented by the adjacency matrices of its component networks.
  • Similarity measures: Interlayer mutual information measures similarity between recurrence structures through the joint degree distribution of two layers.It uses degree sequences rather than the original time series, thereby comparing topological recurrence structure.
  • Similarity measures: Average edge overlap quantifies coherence as the average number of identical edges shared across multiplex layers.Both edge overlap and interlayer mutual information compare linking structures between layers.
  • Weighted projection: Layer-pair similarities can be projected into a weighted M × M network, enabling weighted clustering and shortest-path analyses of interlayer organization.The conversion is computationally efficient and retains interlayer information.
  • Interpretation: High interlayer mutual information, edge overlap, and weighted clustering correspond to periodic systems, whereas chaotic systems show lower values; weighted path length exhibits the opposite pattern.The measures’ discriminative power was demonstrated in coupled-map-lattice and paleoclimate examples.

3.9. Inter-system recurrence networks

Inter-system recurrence networks extend recurrence analysis to multiple dynamical systems by combining within-system and cross-system recurrences. Their cross-transitivity and cross-clustering asymmetries can identify coupling direction, although the result is heuristic and has a scope boundary near synchronization.

  • Cross-recurrence: Cross-recurrence compares delayed close encounters between trajectories from distinct systems sharing a phase space, rather than returns of one system to a prior state.The cross-recurrence matrix uses a prescribed distance threshold and can be asymmetric or non-square for unequal series lengths.
  • Cross-recurrence: Because cross-recurrence connects two distinct vertex groups, its matrix defines a bipartite cross-recurrence network rather than an ordinary single-system recurrence network.Standard network measures may require generalization for this structure.
  • Inter-system recurrence networks: Inter-system recurrence networks combine single-system and cross-recurrence matrices into an undirected, unweighted graph partitioned into within-system and cross-system structures.The construction uses a matrix of within-system thresholds and cross-system distance thresholds.
  • Geometric signatures of coupling: Cross-transitivity and global cross-clustering can detect coupling direction over a wide coupling-strength range using approximately 10^2 to 10^3 samples.The proposed interpretation relies on transitivity-based geometric characteristics and remains heuristic without precise applicability conditions.
  • Geometric signatures of coupling: For unidirectional coupling X → Y, the driven system’s attraction toward the driver increases cross-triangle counts with baseline in X, producing asymmetric cross-network measures.Moderate coupling can also increase the driven system’s dimension and reduce the corresponding cross-transitivity in the opposite orientation.
  • Geometric signatures of coupling: Near and beyond generalized synchronization, cross-clustering measures become statistically indistinguishable because the driven dynamics are locked to the driver.This removes the asymmetry used to identify coupling direction.

3.10. Joint recurrence networks

Joint recurrence networks encode simultaneous recurrences across multiple systems and allow standard network analysis of their combined recurrence structure. Extensions relax simultaneity requirements, while joint-network properties can reveal effective dimensionality and synchronization but become vulnerable as system count or noise increases.

  • Construction: A joint recurrence network connects two time points only when recurrences occur simultaneously in all M time series.Its adjacency matrix is the element-wise product of the individual recurrence matrices and represents an undirected, unweighted graph.
  • Construction: Unlike inter-system recurrence networks, joint recurrence networks require common sampling times and equal series lengths but allow different observables, units, and phase spaces.Vertices are explicitly tied to shared time points.
  • Network interpretation: Comparing joint and single-system recurrence-network properties provides information about neighborhood similarity and the effective degrees of freedom of the combined system.Interdependencies can reduce combined-system degrees of freedom relative to uncoupled components.
  • Limitations: Joint recurrences become less likely as the number of interacting systems increases, and observational noise can further reduce the joint recurrence rate.This limitation can persist even under strong interdependence.
  • f-joint recurrence networks: f-joint recurrence networks relax the strict all-system simultaneity requirement by requiring recurrences in at least a fraction f of systems, with the standard JRN recovered at f = 1.With fixed individual thresholds, the number of f-JRN edges decreases monotonically as f increases.
  • Network properties and synchronization: Joint-recurrence transitivity properties can reveal complex synchronization scenarios, including generalized-synchronization onset, in coupled chaotic oscillators.The approach is presented as promising for interconnections among qualitatively distinct observables in real-world data.
  • Network properties and synchronization: Transitivity-dimension redundancies provide an alternative indicator based on effective dimensions of individual systems.This extends the use of dimensional information for multivariate recurrence analysis.

3.11. Other types of proximity networks

Other proximity networks compare cycles, episodes, or embedded state vectors using correlation, mutual information, phase-space distance, or related similarity measures. These methods avoid explicit time-delay embedding in cycle networks but require careful choices about sampling, thresholds, and embedding dimension.

  • Cycle networks: Cycle networks represent individual oscillatory cycles as vertices and connect them according to a similarity measure such as correlation or phase-space distance.The cycle correlation index compares cycles after optimizing their relative alignment, while thresholding determines network links.
  • Cycle networks: Cycle networks avoid explicit time-delay embedding, tolerate sufficiently small additive noise, and remain invariant under reordering of cycles.These advantages depend on clearly identifying individual cycles from the time series.
  • Practical considerations: Cycle-network construction requires sufficiently high sampling rates because coarse sampling can make even identical cycles appear dissimilar.Both cycle lengths must be reasonably large for reliable correlation or phase-space-distance estimates.
  • Episode networks: Episode networks extend cycle networks by grouping ne ≥1 consecutive cycles, improving mutual-information estimates while introducing an optimizable parameter ne.Episodes address the lower information content of very short cycles, although the method becomes parametric.
  • Applications: Applications include distinguishing affected from healthy cardiac recordings, characterizing gas–liquid flow patterns, and analyzing stock-price return and amplitude series.In cardiac recordings, affected patients showed more variable degree distributions than healthy volunteers, alongside differences in other network measures.
  • Correlation networks: Correlation networks connect embedded state vectors when their Pearson correlation exceeds a threshold, but high embedding dimensions can obscure short-term dynamics.For small embedding dimensions, IOTA provides a permutation-based alternative and can produce directed correlation networks.

4. Visibility graphs

Visibility graphs convert time series into networks using visibility relations, enabling analyses of degree distributions, temporal structure, multivariate coherence, and dynamical properties. Their variants offer useful analytical and practical advantages, but interpretation can be sensitive to process type, sampling uncertainty, missing data, and network-measure choice.

  • Degree distributions: VG degree distributions reflect underlying dynamics, with power-law forms reported for fractal processes and exponential forms for many random processes.For fractal processes, p(k) ∼ k^-γ; for uncorrelated random series, HVGs share an exponential degree distribution regardless of p(x).
  • Distinguishing stochastic and deterministic dynamics: The HVG threshold λc = ln(3/2) was proposed to distinguish correlated stochastic from chaotic dynamics, but negative-coefficient AR(1) processes violate this criterion.For ϕ1 < 0, HVG slopes can be smaller than ln(3/2), so the threshold is not a general separation law.
  • Distinguishing stochastic and deterministic dynamics: Correlated stochastic-process results challenge exponential-degree interpretations, with HVG distributions reported as parabolic exponential forms dependent on the Hurst exponent.Numerical analyses also suggest that VG statistics may not reliably extract time-series correlation information and need not match HVG statistics.
  • Visibility-graph variants: VG and HVG statistics differ systematically: HVG edges form a subset of VG edges, while HVGs provide simpler algorithms and can require longer series because they contain fewer edges.VGs are invariant under affine transformations, whereas HVGs are not; HVG degree sequences can also encode the associated adjacency matrix and symbolically discretize the time series.
  • Practical considerations: Measurement and sampling imperfections alter VG properties: signal-to-noise ratio has considerable effects, while uncertain timings affect local properties but less than observable noise in the reported comparison.Irregular timing can leave p(k) nearly unchanged in one seismic comparison, although network measures remain affected by timing uncertainty.
  • Multivariate and global properties: Multiplex VGs extend visibility analysis to multivariate signals, using edge overlap as a proxy for coherence across variables and restricting analyses mainly to neighborhood-based measures.Higher average edge overlap ω indicates greater microscopic structural correlation, while path-based measures are difficult to decompose into retarded and advanced contributions.

5. Transition networks

Transition networks represent time-series dynamics as directed graphs of discrete states or patterns, with links encoding observed succession and often transition probabilities. Their coarse-grained structure can characterize attractors, dynamical regimes, synchronization transitions, and multivariate relationships, while symbolic encoding and stationarity impose important scope conditions.

  • Construction: Transition networks map discrete states or patterns to nodes and observed succession to directed links, yielding a Markov-chain representation of the dynamics.Weighted graphs retain empirical transition probabilities; unweighted variants retain only pairs with non-zero mutual transition probabilities.
  • Construction: Continuous-valued time series require symbolic encoding before transition-network construction, which compresses detailed dynamics while retaining selected dynamical information.There is no universally optimal encoding; the choice depends on the dynamical features, time scales, and computational resources of interest.
  • Dynamical interpretation: Absorbing and recurrent nodes provide a coarse-grained description of attractors for dissipative dynamics when trajectories remain within a finite phase-space volume.The associated finite-state Markov chain contains states corresponding to phase-space segments of the system’s attractor or attractors.
  • Univariate ordinal patterns: Ordinal pattern transition networks distinguish qualitatively different dynamics: periodic series produce ring structures, whereas chaotic series produce band- or tube-like structures.Mean out-degree and out-degree variance are reported as network measures that can track dynamical differences.
  • Synchronization transitions: During synchronization, transition networks evolve from relatively random transitions among possible patterns toward transitions among fewer patterns, with π3 and π6 forbidden under complete synchronization.For full phase synchronization, p(πq) = 1/6 for π1, π2, π4, π5, π7, and π8, while π3 and π6 are absent.
  • Multivariate networks: Cross and joint ordinal pattern transition networks provide complementary multivariate descriptions, with joint patterns reducing amplitude dependence and extending naturally to more than two subsystems.Generalizing joint networks to three or more coupled subsystems is straightforward, whereas constructing cross networks for three or more subsystems remains challenging.

6. Real-world applications

The reviewed applications show that time-series networks expose dynamical structure across paleoclimate, solar activity, ocean circulation, and physiological data. Recurrence, visibility, and transition-network measures identify regime changes, asymmetries, interdependencies, and complexity patterns, while also revealing setting-specific limitations.

  • Recurrence networks: Inter-system recurrence networks were used to study interdependencies between the Indian and East Asian summer monsoon branches from speleothem oxygen-isotope records.The records came from caves in China and Oman.
  • Recurrence networks: RN motif rankings remained conserved across experimental conditions even as absolute motif frequencies changed with increasing heterogeneity.Classical complexity measures independently confirmed increasing complexity alongside heterogeneous RN motif distributions.
  • Recurrence networks: The 1.1–0.7 Ma B.P. transition appeared at ODP sites 967 and 721/722 but not site 659, showing that recurrence analysis may miss local signatures or subtle events.The authors attribute the absence alternatively to insensitivity to trend changes or insufficient data quality or resolution.
  • Recurrence networks: Recurrence-network transitivity and average path length identified three significant African dust-dynamics transition epochs: 3.5–3.0, 2.25–1.6, and 1.1–0.7 Ma B.P.The epochs were significant in at least two analyzed records.
  • Visibility graphs: Visibility graphs distinguish solar-cycle structure and extrema: strong minima are more temporally clustered than strong maxima, while communities mainly represent consecutive solar cycles.Large-degree hubs can identify solar cycles and connect multiple cycles through high visibility.

7. Software implementation – pyunicorn

The report presents pyunicorn as an open, modular infrastructure combining complex-network and nonlinear-time-series methods. Its broad scope and compiled sparse-data implementations support large analyses, while the authors stress that future uses must remain theoretically well founded.

  • Package overview: pyunicorn unites complex network theory and nonlinear time-series analysis in a performant, modular, and flexible Python package.The package is presented together with its software structure and computational considerations.
  • Package overview: The library provides shared infrastructure for methods developed across participating research groups and is fully open sourced under the BSD 3-Clause license.Its development has incorporated contributions from users worldwide.
  • Architecture: pyunicorn contains five subpackages, including core for general, spatial, multiplex, interacting, and node-weighted network analysis.The passage introduces the package organization and begins describing the subpackages.
  • Performance and applications: The package supports applications in neuroscience, infrastructure, and climatology, with demanding algorithms implemented in fast compiled languages on sparse data structures.These choices target performant analysis of large networks and time-series datasets.
  • Architecture: Its modular object-oriented architecture enables parsimonious combinations of data structures, methods, and algorithms, including recurrence-network analysis from recurrence plots and complex networks.The package is designed to combine methods from different fields.
  • Scope and caution: The authors caution that extensions should be theoretically well founded and motivated by well-posed, relevant research questions.This caveat accompanies pyunicorn’s potential to facilitate methodological development.
  • Related software: The CRP Toolbox offers an alternative MATLAB implementation for recurrence matrices and measures including degree, clustering coefficients, and transitivity.Mathematica demonstrations also cover basic recurrence-network elements.

8. Conclusions and future perspectives

The report reviews recurrence networks, visibility graphs, and transition networks as complementary tools for nonlinear time series analysis. It highlights their insights and applications while identifying open challenges involving nonstationarity, multiple time scales, information recovery, and integration with data mining.

  • Conclusions: The review organizes time series network methods into recurrence networks, visibility graphs, and transition networks.These approaches draw on recurrences, visibility concepts, and Markov chains.
  • Conclusions: Network methods provide alternative measures of system geometry and extend analysis to short, complex, and multivariate time series.They can characterize dynamics, distinguish regimes, identify transitions, test reversibility, and predict future states.
  • Conclusions: Applications have yielded theoretical and practical insights across diverse dynamical systems and scientific fields.The review presents these insights as evidence of the added value of network methods beyond standard linear and nonlinear time series analysis.
  • Future perspectives: Most existing network transformations are designed primarily for stationary systems, although real-world time series often exhibit changing dynamical patterns.Generalizations to account for nonstationarity remain desirable.
  • Future perspectives: Sliding-window evolving networks trace changes in network properties over time but cannot capture all temporal structure in the underlying series.Additional temporal dimensions are needed to represent detailed succession, emergence, and disappearance of network structures.
  • Future perspectives: Multilayer and multiplex network construction must account for phenomena operating across multiple time scales.Scale-sensitive filters such as empirical mode decomposition or wavelet transforms are proposed before network generation, depending on the data and research question.
  • Future perspectives: Future work should combine network approaches with data mining and develop methods that recover or regenerate time series information from network representations.Reconstruction algorithms require empirical parameter choices, and their performance and applicability remain to be evaluated.

Appendix A. Mathematical models

The appendix lists mathematical models used in the paper’s examples, including Lorenz and Rössler systems, autoregressive processes, and the Hénon map. It also specifies coupling and control parameters for selected models.

  • Mathematical models: The appendix identifies the Lorenz system as one of the mathematical models used in the examples.Its listed parameters are r = 28, σ = 10, and β = 8/3.
  • Mathematical models: The Rössler system and diffusively coupled Rössler systems are included among the example models.The appendix refers to coupling through the x component and the second y-component; coupling strength is denoted by µ.
  • Mathematical models: The autoregressive process is specified with real-valued coefficients and Gaussian white-noise errors.The error terms have zero mean and unit variance.
  • Mathematical models: The Hénon map is listed as another model used in the paper’s examples.
  • Parameters and coupling: The appendix notes that symmetric coupling occurs when the two coupling strengths are equal.Specifically, symmetric coupling is achieved if µ12 = µ21.
  • Parameters and coupling: The listed models use coupling strength or a single control parameter to govern selected system configurations.The coupling strength is denoted by µ, while κ denotes a system’s single control parameter.
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