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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

Complex-network methods have had limited overlap with nonlinear time-series analysis despite their practical relevance. This review synthesizes recurrence networks, visibility graphs, and transition networks, showing that they provide valuable theoretical and application-oriented insights into dynamical systems and stochastic processes.

  • Problem

    The report addresses the limited overlap between nonlinear time-series analysis and complex-network methods despite the practical relevance of time-series mining.

    Source: §1 · p. 6

  • Method

    The review examines three main approaches for transforming time series into networks: recurrence networks, visibility graphs, and transition networks.

    Source: §8.1 · p. 105

  • Results

    Complex-network analysis of time series has yielded valuable insights from theoretical dynamical-systems and stochastic-process perspectives and across diverse applications.

    Source: §8.2 · p. 105, §8.1 · p. 105

  • Takeaways & Limitations

    Time-series networks constitute a broad framework for extracting dynamical features through recurrence-, visibility-, and transition-based representations.

    Source: §8.1 · p. 105

    Coarse-grained transition networks require stationarity and ergodicity because estimating transition frequencies over the full series loses detailed temporal information.

    Source: §5.3 · p. 83

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 report reviews how complex network theory can enrich nonlinear time series analysis by transforming temporal data into network representations that reveal dynamical structure. It covers methodological foundations, practical limitations, applications, and future directions across recurrence, visibility, and transition networks.

1.1. Nonlinear time series analysis

  • Nonlinear time series analysis compresses long measurements into characteristic numbers that describe geometry, stability, uncertainty, regime changes, synchronization, and predictability, but remains less established than linear analysis.Its foundations in dynamical systems theory capture nonlinear effects such as sensitive dependence on initial conditions and limited predictability, which classical spectral methods may miss.Source: §1.1 · p. 7
  • Complex network methods can provide more robust estimates of dynamical invariants, including fractal dimensions and Lyapunov exponents, while addressing limitations of nonlinear time series analysis [41].Conventional nonlinear methods face challenges from nonstationarity, embedding choices, finite data, noise, irregular sampling, and computational complexity.Source: §1.1 · p. 8

1.2. Complex network approaches

  • Time series networks extend complex network analysis from multivariate functional connectivity to structural representations of individual time series, revealing dynamical features beyond existing methods.The review emphasizes applications to artificial, observational, and experimental data while distinguishing these approaches from computer-science data-mining tools.Source: §1.2 · pp. 8–9

1.3. Transformations of time series into the complex network domain

  • The review focuses on recurrence networks, visibility graphs, and transition networks, which represent time series through proximity, convexity, and transition probabilities, respectively.These approaches differ in whether edges encode similarity between trajectory segments, visibility among observations, or transitions between discretized states.Source: §1.2 · p. 9, §1.3 · p. 10

1.4. Outline of the report

  • The report proceeds from network theory and vertex-edge definitions to recurrence, visibility, and transition networks, applications, software, and future research directions.It highlights cross-, joint, multiplex, and multivariate extensions, practical considerations, real-world applications, the pyunicorn package [73], and the field’s continued emergence.Source: §1.4 · pp. 11–12

2. Complex network theory

This section introduces graph representations and network measures for characterizing topology, covering connectivity, path properties, global structure, stylized organization, multilayer systems, and interdependent subnetworks. It also presents vertex-level and global cross-network characteristics that quantify proximity, path mediation, connection density, clustering, transitivity, and broader structural properties.

2.1. Basic concepts

  • A complex network is represented as G = (V, E), with vertices and edges that may form directed, undirected, looped, or multiple connections; time-series analyses mainly use simple undirected unweighted graphs.Transition networks are a notable exception receiving special attention as weighted graphs.Source: §2.1 · pp. 12–13
  • Adjacency matrices encode thresholded connections, while paths and path lengths support structural measures whose interpretations and parameter dependence vary with the time-series-to-network mapping.The recurrence-network threshold ε is one example of an algorithmic parameter.Source: §2.1 · p. 13

2.2.1. Vertex characteristics

  • Vertex measures quantify local, meso-scale, and global connectivity through degree, degree distributions, entropy, clustering, closeness, efficiency, and betweenness centrality.Closeness and local efficiency describe geometric centrality, whereas betweenness characterizes importance for information propagation.Source: §2.2.1 · pp. 13–14

2.2.2. Edge characteristics

  • Edge and motif measures characterize neighborhood overlap, shortest-path traversal, and small local connection patterns, with motif frequencies serving as sensitive indicators of structures reconstructed from time series.Motifs typically contain 3 or 4 vertices and are particularly useful for directed networks.Source: §2.2.2 · pp. 14–15

2.2.3. Global network characteristics

  • Global measures summarize edge density, clustering, path length, efficiency, and diameter, while network transitivity corrects clustering interpretations that can be underestimated in heterogeneous degree distributions.The diameter is the maximum shortest-path length and represents network size.Source: §2.2.3 · p. 15

2.3. Stylized facts of complex networks

  • Stylized network organization includes Erdős–Rényi random graphs with Poisson degree distributions, small-world networks combining short paths and high clustering, scale-free networks with heavy-tailed degrees, and assortative or disassortative degree correlations.The Watts–Strogatz model interpolates between ordered lattices and random graphs, while the Barabási–Albert model uses growth and preferential attachment.Source: §2.3 · p. 16

2.4. Multiplex and multilayer networks

  • Multilayer networks represent intra- and inter-layer connections among graph layers, while multiplex networks share one vertex set across layers and allow different edge types.These frameworks address multiscale, multicomponent interactions such as disease spreading, diffusion, and cooperation dynamics.Source: §2.4 · pp. 16–17

2.5.1. Preliminaries

  • For interdependent time-series subnetworks, cross-degree, cross-clustering, cross-closeness, and cross-betweenness generalize graph measures to quantify vertex-level and global interconnectivity.Cross-degree counts direct neighbors in another subnetwork, while cross-clustering measures whether those neighbors are mutually connected.Source: §2.5.1 · pp. 17–18, §2.5.2 · p. 18

2.5.2. Vertex characteristics

  • Vertex characteristics generalize closeness, efficiency, and betweenness to coupled subnetworks, including paths that connect vertices across different subnetworks.Cross-betweenness allows the intermediary vertex to belong to any subnetwork because shortest paths between two subnetworks may traverse other subnetworks.Source: §2.5.2 · p. 19

2.5.3. Global characteristics

  • Global characteristics include cross-edge density, cross-clustering, cross-transitivity, cross-average path length, and global cross-efficiency for quantifying coupled-network structure.Cross-edge density is symmetric for undirected networks, while cross-clustering and cross-transitivity are generally asymmetric; shortest-path measures are symmetric by definition.Source: §2.5.3 · pp. 19–20
  • Additional coupled-network measures can adapt edge, pairwise, global, mesoscopic, and diffusion-based characteristics such as assortativity, motifs, eigenvector centrality, and random-walk betweenness.The explicitly discussed measures reflect characteristics previously used to study interdependence between complex networks.Source: §2.5.3 · p. 20

3. Recurrence networks in phase space · 3.1. Theoretical background · 3.2. Types of recurrence networks

This section develops recurrence networks as phase-space geometric representations of time-series recurrences, beginning with attractor reconstruction and recurrence plots before introducing network variants and extensions. It emphasizes how network structure reveals geometric organization while distinguishing graph paths and measures from dynamical trajectories and recurrence-based analyses.

3. Recurrence networks in phase space

  • Recurrence networks reinterpret phase-space recurrences as complex-network structure, revealing geometric characteristics of the underlying dynamics beyond recurrence quantification, recurrence-time statistics, and invariant estimation.The section also introduces bi- and multivariate extensions based on cross-, joint-, and multiplex recurrence plots.Source: §3 · p. 20

3.1.1. Phase space and attractor reconstruction

  • Time-delay embedding reconstructs m-dimensional state vectors x⃗_i=(x_i,x_i−τ,…,x_i−(m−1)τ), with topological equivalence guaranteed for deterministic systems when m>2D_F.In practice, m and τ can be selected using false nearest neighbors, autocorrelation, or delayed mutual information, and embedding dependence should be checked for experimental data.Source: §3.1.1 · pp. 20–21

3.1.2. Recurrences and recurrence plots

  • A recurrence plot marks pairs of phase-space states as recurrent when their distance under a chosen norm is below threshold ε, producing a binary matrix and visual representation of recurrence structure.Recurrence plots support analyses of dynamical patterns, while the choice of ε affects the resulting plot and statistics.Source: §3.1.2 · p. 21
  • Recurrence-plot methods include RQA measures of point, diagonal, and vertical structures, estimates of dynamical invariants, and analyses of interrelations, coupling, and phase synchronization [120].Variants such as fuzzy recurrence plots and data-adaptive phase partitions target clearer or more specific recurrence domains.Source: §3.1.2 · p. 22

3.2. Types of recurrence networks

  • An ε-recurrence network treats sampled state vectors as vertices connected by undirected edges when they are mutually within ε, yielding a simple graph whose adjacency matrix inherits recurrence-matrix symmetry.Network paths represent sequences of overlapping ε-neighborhoods and are distinct from the trajectory describing dynamical evolution.Source: §3.2.1 · pp. 22–23

3.2.4. Algorithmic variants

  • Algorithmic variants include weighted or unthresholded networks, dimension-based complexity indices, symbolic and frequency-degree mappings, fuzzy or grammatical recurrence representations, and windowed or coarse-grained methods for long series.The standard RN computation scales as O(N^2), motivating sliding windows, segmentation, and meta-recurrence constructions, while some proposed complexity indices lack demonstrated advantages over classical fixed-embedding measures.Source: §3.2.1 · p. 23, §3.2.4 · p. 25

3.2.2. k-nearest neighbor networks

  • k-nearest-neighbor networks replace fixed-distance neighborhoods with fixed neighbor counts, generally producing directed graphs whose unidirectional-edge density relates to gradients of invariant density.Adaptive nearest-neighbor networks symmetrize these relationships and use motif rankings to discriminate dynamical types across turbulence, music, stochastic, and multifractal data [54] [129].Source: §3.2.2 · p. 24, §3.2.3 · p. 24

3.2.3. Adaptive neighbor networks

  • Adaptive-neighbor motif prevalence is strongly dependent on local attractor dimension, with comparable superfamily effects appearing even for randomly distributed states in spaces of varying dimensions [130].No fundamental theory fully explains these empirical superfamily phenomena; the dimensionality-based account remains heuristic.Source: §3.2.3 · p. 24

3.3. Complex network characteristics of RN

Recurrence networks translate recurrence matrices into geometric network representations of attractors, complementing dynamical recurrence-plot analysis by discarding temporal ordering. Their characteristics quantify local, edge-based, and global phase-space structure at a chosen recurrence threshold.

3.3. Complex network characteristics of RN

  • RN analysis uses complex-network measures to characterize geometric phase-space structure, whereas recurrence-plot analysis explicitly considers dynamical aspects, including sequences of state vectors [3, 4, 22, 42, 58].Source: §3.3 · pp. 25–26

3.3.1. Vertex characteristics

  • Vertex measures map attractor geometry to local properties: degree density estimates state density, clustering reflects geometric alignment near low-period unstable periodic orbits, and betweenness identifies fragmentation bottlenecks.Closeness and local efficiency are highest for vertices near the network center, while clustering bands in the Lorenz system correspond to known low-periodic orbits.Source: §3.3.1 · pp. 27–28

3.3.2. Edge characteristics

  • Edge characteristics quantify neighborhood similarity and phase-space fragmentation, with matching index and edge betweenness showing a clear anti-correlation when both are non-zero [53].Large matching index indicates closely located states that can exchange as shortest-path members, producing lower edge betweenness.Source: §3.3.2 · p. 28

3.3.3. Global network characteristics

  • Global RN measures connect network statistics to attractor geometry: edge density equals recurrence rate, clustering and transitivity proxy local and global dimensionality, and average path length reflects phase-space separation.Edge density increases with recurrence threshold, whereas average path length decreases because it measures geodesic distances in units of the threshold ε.Source: §3.3.3 · p. 28

3.4. Analytical theory of RN

The analytical theory of recurrence networks (RNs) exploits their equivalence to random geometric graphs (RGGs) to interpret network measures as continuous geometric properties of chaotic attractors. In the continuum limit, RN shortest paths converge to attractor geodesics, while discrete measures approximate density-, geometry-, and geodesic-based quantities.

3.4. Analytical theory of RN

  • RNs are hard-threshold RGGs whose vertex density is the dynamical system’s invariant density, so RN characteristics become discrete approximations of continuous geometric properties of the attractor [145].This correspondence links local and global transitivity measures to the local and global fractal dimensions of the attractor, respectively.Source: §3.4.1 · p. 29, §3.4 · p. 29, §3.4.2 · p. 29

3.4.1. Preliminaries: random geometric graphs

  • Sampling can distort RGG/RN spatial coverage through attractor-frequency synchronization, whereas ergodic systems yield equivalent large-N properties from long trajectories, independent realizations, or invariant-density samples.Bootstrapping available state vectors is often preferred to regular trajectory sampling in practice.Source: §3.4.1 · p. 29

3.4.2. Analytical description of ε-recurrence networks

  • In the limits ε →0 and N →∞, RN shortest paths approximate global geodesics on the invariant set S, independently of the chosen metric [145].This provides the continuum analog of network shortest-path lengths and underpins path-based measures such as closeness, betweenness, efficiency, and average path length.Source: §3.4.2 · p. 30
  • The continuous framework defines analogs of RN degree, clustering, centrality, matching, density, transitivity, path-length, and efficiency measures through probability and geodesic functionals on S.Their discrete estimators are the corresponding standard RN measures; further extensions include assortativity, diameter, radius, motifs, eigenvector centrality, and random-walk betweenness [145].Source: §3.4.2 · p. 31

3.5. General properties of recurrence networks

Recurrence networks have analytically characterized geometric and topological properties, with transitivity providing a normalized measure of attractor dimensionality and dynamical complexity. Their degree distributions, path lengths, mixing patterns, and local clustering depend on invariant-density structure, recurrence threshold, sampling, and network size.

3.5.1. Degree distributions of RNs

  • Under ergodicity, transient-free sampling, and sampling intervals co-prime to system periods, RN vertices behave as random samples from the attractor’s invariant density, yielding an analytical degree distribution.For large N, binomial node degrees are approximated by Poisson distributions whose parameter depends on the local invariant density.Source: §3.5.1 · p. 33
  • Power-law RN degree distributions arise when the invariant density has power-law singularities, but sufficiently large ε masks this scaling and no general necessary-and-sufficient conditions are known for higher-dimensional systems.The scaling exponent is not generally equivalent to the attractor’s fractal dimension because it reflects local invariant-density shape rather than global fractal structure.Source: §3.5.1 · p. 34

3.5.2. Small-world effect

  • RNs do not intrinsically exhibit small-world effects: for fixed ε, average path lengths remain bounded independently of N and scale as ε^-1 rather than the usual log N.Noise or thresholds comparable to attractor size can nevertheless produce numerically observed small-world properties.Source: §3.5.2 · pp. 34–35

3.5.3. Assortative vs. disassortative mixing

  • RNs generally show assortative mixing when the invariant density is continuous or differentiable, while local clustering can characterize point-wise attractor dimension and potentially locate weakly unstable periodic orbits.Clustering-based dimensionality is affected by sharp boundaries and tips, and UPO signatures are limited by ε and N and have not yet been systematically used for localization.Source: §3.5.3 · p. 35, §3.5.5 · p. 38

3.5.4. Path-based characteristics

  • Average path length and transitivity are the most useful global RN measures for dynamical complexity, but path length is unnormalized, ε-dependent, and behaves differently for periodic and chaotic discrete versus continuous systems.For discrete periodic maps, disconnected fully connected components require redefining the average over mutually reachable pairs, whereas continuous-time periodic orbits can have higher path lengths than chaotic attractors.Source: §3.5.4 · pp. 35–36

3.5.5. Dimension characteristics by clustering and transitivity

  • RN transitivity provides a normalized, theoretically grounded measure of effective attractor dimension and can discriminate high from low dynamical complexity across discrete and continuous systems.Its estimates require sufficiently appropriate recurrence thresholds and network sizes; excessive or insufficient thresholds can mask structure or disconnect the network.Source: §3.5.5 · p. 37

3.6. Practical considerations

Practical RN analysis depends critically on selecting recurrence thresholds and embedding parameters that preserve attractor geometry while maintaining connectivity. Noise and uncertainty can alter RN structure and interpretation, requiring adaptive criteria, robustness checks, and transformations for non-stationary data.

3.6.1. Choice of recurrence rate or threshold

  • Recurrence thresholds should generally yield sparse, connected RNs, with ρ ≲0.05 often recommended, while turning-point heuristics are unreliable and may require adaptive or surrogate-assisted selection.Thresholds just above the percolation threshold can preserve fine attractor structure while ensuring a giant component; connectivity-based criteria may fail for disjoint phase-space supports.Source: §3.6.1 · pp. 39–40

3.6.2. Dependence on embedding parameters

  • Embedding choices are problematic for non-stationary processes: finite m and τ can produce spurious or non-unique RN properties, so detrending, deseasonalization, or differencing may be required.For fractional Brownian motion, differencing transforms the process into stationary fractional Gaussian noise, whereas RN interpretations for the original non-stationary process are limited.Source: §3.6.2 · p. 41

3.6.3. ε-dependence of RN properties

  • As ε increases, average path length decreases approximately as 1/ε, whereas clustering generally increases approximately linearly at intermediate thresholds but depends on the dynamical system.Very small thresholds can fragment finite-sample RNs into disconnected clusters, limiting the interpretation of network measures.Source: §3.6.3 · p. 42, Figure 10 · p. 42

3.6.4. Stability and robustness against noise

  • Weak observational noise is often addressed with ε ∼5σ, but noise above 40% impairs discrimination for clustering, above 20% for path length, and at 50% or more drives RN structure toward random geometric graphs.Under colored noise, clustering is not a reliable discriminator, whereas shortest path length remains effective for both white and colored noise; interpretation depends on the tested hypothesis.Source: §3.6.4 · pp. 42–43
  • Measurement errors, noise, and irregular sampling make binary recurrence decisions uncertain, motivating recurrence-probability methods that model observational uncertainty and improve the significance of detecting abrupt transitions.These methods currently estimate recurrence probabilities without embedding because meaningful embedding of time series of probability densities is unavailable.Source: §3.6.4 · p. 43

3.7. Numerical examples

The numerical examples show that recurrence-network measures distinguish dynamical regimes across dissipative, Hamiltonian, non-stationary, and long-range-correlated stochastic systems. Their interpretation depends on suitable sampling, embedding, and recurrence settings, especially for stochastic and non-stationary processes.

3.7. Numerical examples

  • Across numerical examples, local and global RN characteristics visualize attractor geometry, identify dynamical transitions, and track changing complexity in stationary and drifting-parameter systems [52] [53] [54] [152] [160].Applications include vertex-based phase-space coloring and global measures such as transitivity and average path length.Source: §3.7 · p. 43

3.7.1. Parameter space in the R¨ossler system

  • RN transitivity and average path length distinguish periodic from chaotic Rössler solutions, with periodic windows showing values near T̂ = 0.75 and higher T̂ and L̂ than chaos [152].These measures reveal sequences of periodic–chaotic transitions in a two-dimensional parameter-space section displaying self-similar “shrimp” structures.Source: §3.7.1 · p. 44, Figure 11 · p. 44

3.7.2. Hamiltonian dynamics

  • RN analysis reliably detects sticky regular organization in standard-map dynamics from relatively short series, while fixed recurrence rate captures intermingled regular and chaotic regions [176].Quasi-periodic trajectories have larger T̂ and Ĉ than filling chaotic trajectories, whereas L̂ remains strongly affected by orbit size unless recurrence rate is fixed.Source: §3.7.2 · pp. 44–45

3.7.3. Non-stationary deterministic systems

  • Sliding-window RN measures recover bifurcation locations in linearly drifting logistic-map and Rössler systems, provided the recurrence networks remain sufficiently large for reliable statistics.The estimated transitions coincide well with autonomous-system bifurcation points, indicating limited transient influence near those points in the considered examples.Source: §3.7.3 · p. 45

3.7.4. Long-range correlated stochastic systems

  • For fGn with H > 0.5, transitivity and clustering depend little on H at fixed embedding dimension, whereas for H < 0.5 both rise as H decreases because τ = 1 is suboptimal.Non-stationary fBm requires transformation to stationary increments, and embedding choices remain constrained by finite sample size and the theoretically infinite stochastic embedding dimension.Source: §3.7.4 · pp. 45–46

3.8. Multiplex recurrence networks

Multiplex recurrence networks extend recurrence-network analysis to multivariate time series by assigning each component to a time-aligned layer. Interlayer similarity, coherence, and weighted projections characterize shared recurrence structures and distinguish periodic from chaotic dynamics.

3.8. Multiplex recurrence networks

  • Multiplex recurrence networks construct one recurrence-network layer per component of an M-dimensional time series, using shared time-indexed nodes and interlayer links only between temporal counterparts.All component time series must have the same time points, so each node is uniquely associated with a common time index across layers.Source: §3.8 · p. 47
  • Interlayer mutual information I_αβ measures similarity in degree-based recurrence structures, while average edge overlap ω quantifies coherence through links shared across layers.Both measures compare layers pairwise; I_αβ uses degree sequences rather than the original time series, whereas ω averages identical edges across layers.Source: §3.8 · p. 48
  • The multiplex can be projected into an M × M weighted network whose layer-to-layer edges encode I_αβ, enabling efficient computation of weighted clustering Cw and path length Lw.This projection retains interlayer information while permitting traditional weighted-network characterization.Source: §3.8 · p. 48
  • High I_αβ, ω, and Cw values characterize periodic systems, whereas lower values indicate more chaotic systems; Lw shows the opposite trend because denser networks have shorter diameters.These measures discriminated periodic and chaotic behavior in coupled map lattices and real-world paleoclimate time series.Source: §3.8 · p. 48

3.9. Inter-system recurrence networks

Inter-system recurrence networks extend recurrence analysis from individual systems to coupled multivariate dynamics by combining cross-recurrence and single-system recurrence structures. Their geometric measures can reveal coupling direction, although detection depends on coupling strength, sampling, network size, and shared phase-space requirements.

3.9.1. From cross-recurrence plots to cross-recurrence networks

  • Cross-recurrence matrices encode delayed close encounters between two systems as bipartite edges, rather than returns of one isolated system to a previous state.The matrix may be asymmetric and non-square when systems differ in trajectory geometry or series length, so it cannot directly serve as a single-system recurrence-network adjacency matrix.Source: §3.9.1 · p. 49

3.9.2. Coupled networks framework for M subsystems

  • Inter-system recurrence networks combine within-system recurrence matrices and cross-recurrence matrices into an undirected, unweighted network of subnetworks for M systems.Vertices represent phase-space state vectors, while within- and between-system edges represent mutual closeness; thresholds may be equal or selected to equalize within-system recurrence rates and reduce cross-recurrence densities.Source: §3.9.2 · pp. 50–51
  • IRN construction requires time series sharing a common phase space and physical units, but permits arbitrary sampling times and different lengths because it uses phase-space neighborhoods rather than temporal ordering.Equal thresholds can reinterpret the IRN as a recurrence network of concatenated data, although this obscures the natural partition and leaves community-based interpretation unresolved.Source: §3.9.2 · pp. 50–51

3.9.3. Analytical description

  • The analytical framework defines local and global cross-measures that separate overlap and density effects from global path geometry, including cross-clustering, cross-transitivity, cross-efficiency, and cross-betweenness.Cross-clustering and cross-transitivity are generally asymmetric, whereas path-based cross-average path length and global cross-efficiency are invariant under exchanging the two attractor sets.Source: §3.9.3 · pp. 51–53

3.9.4. Geometric signatures of coupling

  • Global cross-clustering coefficients and cross-transitivity can identify unidirectional coupling over moderate strengths, but become indistinguishable near synchronization or at very weak coupling.For X → Y, the expected asymmetry is ˆC_YX > ˆC_XY and ˆT_YX > ˆT_XY; Fig. 16 reports this using 200 realizations with N = 1,500 points per system.Source: §3.9.4 · pp. 53–54, Figure 16 · p. 54

3.10. Joint recurrence networks

Joint recurrence networks combine simultaneous recurrences across multiple systems, enabling network-based analysis of interdependence without requiring identical observables or phase spaces. Their properties can reveal synchronization and effective dimensional relationships, while f-JRNs relax strict simultaneity to improve robustness under noise.

3.10.2. Network interpretation

  • A joint recurrence network is the element-wise product of individual recurrence networks, so each edge represents simultaneous recurrence across all systems.The graph is undirected and unweighted, with vertices tied to common time points; componentwise logical AND gives the same construction.Source: §3.10.2 · p. 55
  • JRNs compare recurrences rather than state vectors, allowing systems with different physical quantities, units, and phase spaces while exposing effective degrees-of-freedom reduction from interdependence.Comparing JRN and single-system RN properties probes neighborhood similarity between the combined phase space and individual-system projections.Source: §3.10.2 · p. 55
  • f-JRNs require recurrences in at least a fraction f of systems, generalizing standard JRNs and yielding fewer edges as f increases, but remain largely unexplored.This relaxation is intended to address the decreasing likelihood of joint recurrence with more systems and observational noise, with applications left for future research.Source: §3.10.2 · pp. 55–56

3.10.3. Network properties and synchronization

  • For generalized synchronization, JPR compares joint with equal individual recurrence rates and should approach 1, while higher-order transitivity and effective-dimension redundancy offer alternatives with limitations [192].JPR significance testing usually requires complex surrogates, and observational noise can prevent reaching the expected optimum JPR = 1; transitivity-based indicators lack rigorous theoretical justification.Source: §3.10.3 · pp. 57–58
  • DLI approaches one at generalized-synchronization onset but remains about 0.2 or lower without synchronization, while driven-system transitivity decreases before onset.Ensemble simulations of coupled Rössler systems showed a marked DLI increase at synchronization and systematic pre-onset changes in the driven system.Source: §3.10.3 · pp. 58–59, Figure 17 · p. 59

3.11. Other types of proximity networks

The section reviews cycle networks, which represent oscillatory cycles or multi-cycle episodes as nodes linked by trajectory similarity, and correlation networks, which connect highly correlated embedded state vectors. These approaches avoid or extend standard recurrence-network constructions but depend on appropriate sampling, embedding, or similarity estimation choices.

3.11.1. Cycle networks

  • Cycle networks represent individual cycles as vertices and connect pairs whose trajectory segments are sufficiently similar, using measures such as shifted cross-correlation or phase-space distance [49, 214, 217].The cycle correlation index maximizes cross-correlation while sliding the shorter cycle against the longer one; alternative similarity measures are also possible.Source: §3.11.1 · pp. 59–60
  • Cycle networks avoid explicit time-delay embedding, tolerate sufficiently small additive noise, and yield structured networks for chaotic nonlinear systems versus more random networks for linear or periodic systems.Network clustering can also help locate unstable periodic orbits because their vertices form clusters in the cycle network.Source: §3.11.1 · p. 60
  • Reliable cycle-network construction requires clearly identifiable cycles and sufficiently high sampling rates, because coarse sampling can miss links between cycles that are close in phase space.Peak- or trough-based cycle segmentation is preferred, while correlation indices are useful for noisy or non-stationary data and phase-space distance is physically more meaningful.Source: §3.11.1 · p. 60
  • Episode networks extend cycle networks by grouping ne consecutive cycles and using mutual information, improving statistical estimation while introducing ne as an optimizable parameter.Very short cycles convey less information, motivating episodes despite the parameter-free definition of a cycle.Source: §3.11.1 · pp. 60–61

3.11.2. Correlation networks

  • Correlation networks connect embedded state vectors when their Pearson correlation exceeds a threshold, but high embedding dimensions can obscure short-term dynamics and induce spurious correlations.For short embeddings such as m = 10, IOTA provides a more statistically robust, directed alternative that has been applied to short-term heartbeat series.Source: §3.11.2 · pp. 61–62

4. Visibility graphs

This section develops visibility-graph methods for representing scalar time series as geometric networks and characterizing their structural and temporal properties. It covers major graph variants, global and directed measures, multivariate extensions, diagnostic uses, and limitations arising from data quality, sampling, finite size, and boundary effects.

4. Visibility graphs

  • VGs represent scalar time series as networks whose vertices are observations and whose edges encode geometric or ordering-based visibility relationships.The approach provides a combinatorial representation of the underlying dynamical system and includes natural, horizontal, and related variants.Source: §4 · p. 62, §4.1 · p. 62

4.1.1. Natural visibility graphs

  • Natural VGs preserve temporal information, always connect consecutive observations, avoid algorithmic parameter choices, and have quadratic basic time complexity that can be reduced to approximately O(N log N).The faster transform improves the natural VG algorithm, while HVG construction already reaches approximately O(N) complexity.Source: §4.1.1 · p. 62, §4.1.3 · p. 65

4.1.2. Horizontal visibility graphs

  • HVGs impose a stricter visibility criterion than standard VGs, yielding a sparser edge set, greater algorithmic simplicity, and analytically tractable properties, but requiring longer records in some applications.Standard VGs are affine-invariant, whereas HVGs are not; HVG sparsity increases data-length demands for tests such as time-reversal asymmetry.Source: §4.1.2 · p. 63

4.1.3. Other variants of (H)VG

  • Generalized (H)VGs add direction, distance- and amplitude-dependent weights, penetrability, viewing-angle parameters, multiscale constructions, and graphlet or attribute-based analyses for specialized dynamical signals.These variants support time-reversal analysis, characterize recurrent neural-network dynamics, analyze EEG and electromechanical signals, track bifurcations, and perform pattern recognition.Source: §4.1.3 · p. 64

4.2.1. Degree distributions

  • Degree distributions distinguish broad process classes: periodic signals produce regular motifs, white noise produces exponential structure, and fractal processes often exhibit power-law behavior linked heuristically to Hurst exponents.For fractal records, direct analysis of the degree sequence with detrended fluctuation analysis can quantify multifractality better than standard VG degree-distribution analysis; inversion of the series exposes strong minima.Source: §4.2.1 · pp. 65–66

4.2.2. Distinguishing stochastic and deterministic dynamics

  • The proposed critical slope λc = ln(3/2) is not a general separator between stochastic and chaotic dynamics, because negatively correlated AR(1) processes can have λ < λc and fractional-process results challenge simple exponential interpretations.For positively correlated AR(1) processes, VG and HVG exponents vary oppositely with ϕ1, while HVG λ increases and remains above λc as correlation length grows.Source: §4.2.2 · pp. 66–67

4.2.3. Degree sequences of horizontal visibility graphs

  • The HVG degree sequence can encode the underlying dynamics without information loss: its block entropies converge toward the system’s Kolmogorov-Sinai entropy, and it is bijective with the adjacency matrix.This makes the degree sequence a symbolic discretization and asymptotically a generating partition of the system.Source: §4.2.3 · p. 67

4.2.4. Local network properties

  • VG centrality measures have distinct structural meanings but are strongly affected by record boundaries, which can systematically underestimate degree and other centralities near the ends of short time series.Clustering reflects hierarchical or modular organization, whereas betweenness and closeness also mix local visibility with temporal position; edge density ranges from maximal for globally convex series to low values for irregular fluctuations.Source: §4.2.4 · pp. 67–68, §4.2.5 · p. 68

4.2.5. Global network properties

  • Visibility-graph global measures connect clustering, path length, assortativity, and motifs to small-world, fractal, and dynamical properties.For fBm, C decreases with H, while L increases exponentially with H; degree correlations distinguish non-fractal fBm graphs from fractal Conway-series graphs [276].Source: §4.2.5 · pp. 68–69

4.3. Practical considerations

  • Experimental (H)VG analysis remains insufficiently validated because missing data, heteroscedastic noise, and uncertain sampling times can alter network properties.Existing practical studies mainly use idealized model data, and their effects on real-world networks remain incompletely addressed.Source: §4.3 · p. 69

4.3.1. Missing data

  • Visibility graphs tolerate nonuniform sampling, but missing-value handling can substantially change network structure, especially when missing observations remove hubs or fragment the graph.Random removal notably affects global and local properties, while ignoring missing values primarily changes closeness distributions in the reported KS analysis.Source: §4.3.1 · p. 70

4.3.2. Homo- and heteroscedastic uncertainties

  • Additive and multiplicative measurement noise systematically shift vertex-property distributions toward those of the noise process as signal-to-noise ratio decreases.The reported saturation of KS statistics for moderate noise reflects variability among independent realizations of the same Gaussian signal process.Source: §4.3.2 · pp. 70–71

4.3.3. Uneven and irregular timings

  • Uncertain sampling times affect local vertex-property distributions, although their impact is smaller than measurement noise and irregular timing may leave degree distributions nearly unchanged.For seismic series, replacing random event times with regular times produced almost identical p(k), whereas timing uncertainty still altered network measures.Source: §4.3.3 · pp. 71–72, Figure 22 · p. 72

4.4. Multivariate visibility graph methods

  • Multivariate visibility methods extend univariate graphs through multiplex, cross-, and joint constructions for describing coupling, shared structure, and information transfer across variables.Multiplex graphs map each variable to an HVG layer and connect multivariate analysis with multilayer-network descriptors.Source: §4.4 · p. 72, §4.4.1 · pp. 72–73

4.4.1. Multiplex visibility graphs

  • Multiplex edge overlap and interlayer degree mutual information quantify structural coherence and information flow among variables, while joint and excess degrees capture bivariate asymmetries.Higher ω indicates greater microscopic similarity across layers, and ∆ki and ∆relki compare visibility connections between two series [188].Source: §4.4.1 · p. 73, §4.4.2 · p. 74

4.4.3. Visibility graph similarity

  • Visibility-graph similarity combines time-delay embedding, distance-series graphs, and degree-sequence correlation to detect generalized synchronization and represent brain activity.Directed graphs separately encode past and future connections through in- and out-degrees, which are interchanged under time reversal [63].Source: §4.4.3 · p. 74, §4.5.1 · p. 75

4.5.1. Time-directed visibility graphs and characterizations

  • Time-directed graph properties can be decomposed into retarded and advanced degrees or clustering coefficients, but path-based measures resist this decomposition because they depend on global paths [63, 64].Separating forward and backward edge sets offers an alternative that permits path-based measures to be computed independently, although neighborhood-based measures are the primary focus.Source: §4.5.1 · p. 75

4.5.2. Tests for time series irreversibility

  • Directed (H)VGs assess irreversibility by comparing retarded and advanced properties; Kullback-Leibler divergence is zero exactly when in- and out-degree distributions coincide and positive otherwise [64].For stationary series, reversible dynamics require equivalent forward and backward statistics, while statistically significant differences indicate irreversibility.Source: §4.5.2 · pp. 76–78
  • Persistence and anti-persistence both increase D_kld, with positive correlations producing a somewhat stronger effect than negative correlations.The divergence therefore provides a first-approximation measure of time asymmetry for processes with tunable long-range dependence.Source: §4.5.2 · p. 77
  • VG irreversibility tests distinguish reversible white noise from irreversible chaotic dynamics: D_kld(in∥out) vanishes as 1/N for white noise but converges to a finite value for the logistic map [64].The method also correctly distinguishes reversible and irreversible stochastic processes, conservative and dissipative chaotic maps, and noisy dissipative maps.Source: §4.5.2 · pp. 76–77, Figure 23 · p. 77
  • VG-based irreversibility tests avoid ad hoc symbolization and surrogate-data construction, work directly on real-valued and non-uniformly sampled data, but remain sensitive to process details, boundary effects, missing data, noise, sampling, and vertex correlations [63, 296, 309–311].Finite series produce nonzero irreversibility values even for reversible processes, and path-based measures are especially vulnerable to boundary effects.Source: §4.5.2 · pp. 77–79
  • The KS tests nearly never reject reversibility for AR(1), whereas they nearly always reject it for the Hénon map using degrees and always using local clustering coefficients [63].For AR(1), clustering-based rejections remain below the expected 5% false-rejection rate.Source: §4.5.2 · p. 78

5. Transition networks

This section examines transition networks that represent temporal succession among discrete states or patterns as directed graphs, including symbolic encodings for continuous and multivariate data. It covers variants that reveal dynamical structure, synchronization regimes, and complementary transition information, while addressing their dependence on discretization and potential loss of temporal information.

5. Transition networks

  • Transition networks represent states or patterns as nodes and nonzero successive-transition probabilities as directed links, yielding weighted Markov-chain graphs.Unweighted representations omit probabilities, while thresholding can exclude rare transitions.Source: §5 · pp. 79–80

5.1. Symbolic encoding

  • Symbolic encoding offers coarse-graining strategies based on amplitudes, differences, order patterns, or ordinal permutations, with no universally optimal choice for the dynamics or timescales of interest.Proper encodings can preserve essential dynamical information while addressing noise and multistability, but estimated statistics depend on encoding parameters.Source: §5.1 · p. 81

5.2. Markov chains

  • Markov-chain transition matrices encode one-step succession and provide network interpretations of absorbing states, recurrent loops, stationary densities, and eigenvector centrality, although memorylessness may miss longer-term serial correlations.The stationary density is the eigenvector of W with eigenvalue 1 for non-degenerated chains.Source: §5.2 · p. 82

5.3. Coarse-graining based transition networks

  • Coarse-grained transition networks identify absorbing and recurrent states that describe attractors, while temporal unfolding can recover memory effects lost by static representations.Static networks require stationarity and ergodicity; consecutive memory networks and memory entropy distinguish white noise, 1/f noise, AR, periodic, and chaotic series.Source: §5.3 · p. 83

5.3.1. Ordinal pattern transition networks for univariate time series

  • Ordinal pattern transition networks distinguish periodic ring structures from chaotic band or tube-like structures and support dynamical discrimination and change-point detection [41].Their mean out-degree and out-degree variance can track dynamical changes comparably to the largest Lyapunov exponent [41].Source: §5.3.1 · p. 84

5.3.2. Order pattern transition networks for multivariate time series

  • Multivariate ordinal transition networks encode combinations of component-wise increment signs, producing an alphabet of size 2^n that captures dynamics in the associated velocity space.For three-dimensional series, the construction yields K = 8 order patterns; its delay has a different interpretation from classical time-delay embedding.Source: §5.3.2 · pp. 85–86
  • Multivariate order-pattern transition networks can remove self-loops and quantify node and transitional complexity using entropy measures derived from pattern occurrence and transition frequencies.SO corresponds to permutation entropy, while ST measures ordinal-pattern transitional complexity with a normalization differing slightly from earlier definitions.Source: §5.3.2 · pp. 86–87

5.3.3. Ordinal pattern transition networks for synchronization transitions

  • Synchronization regimes are distinguished by ordinal-pattern frequencies, entropy trends, and rotation frequencies as three coupled Rössler oscillators transition from asynchrony to partial and complete phase locking.Below μc,1 = 0.036, ST decreases while SO remains nearly constant; between 0.036 and 0.077, oscillators 1 and 2 lock; above μc,2 = 0.077, all lock and π3 and π6 become forbidden.Source: §5.3.3 · p. 87, Figure 27 · p. 88, Figure 28 · p. 88

5.4. Cross and joint ordinal transition networks

  • Cross and joint ordinal transition networks provide complementary coupling descriptions: COPTNs compare componentwise increment differences, whereas JOPTNs compare increment-product signs and largely disregard amplitudes.JOPTNs generalize straightforwardly to more than two subsystems, while constructing COPTNs for three or more systems remains challenging; local normalization can also cause numerical problems near component roots.Source: §5.4 · pp. 89–90

5.5. Other related approaches

  • Additional symbolic-transition methods characterize cardiac regulation, stock-market volatility, correlation-mode transmission, delayed interactions, and gas–fluid-flow dynamics through tailored coarse-grained or correlation-based encodings.These approaches use RR-interval increments, four volatility states, regression patterns, reduced autoregressive models, or rank-ordered pairwise correlations to trace clinically or dynamically relevant network structure.Source: §5.5 · pp. 90–91
  • Related transition networks extend symbolic representations by combining ordinal patterns with binned amplitudes, thereby retaining information useful for detecting weak nonlinearities.Each node combines an amplitude bin with an ordinal pattern, and directed links connect successive symbol pairs; the approach has also been combined with recurrence and surrogate networks.Source: §5.5 · p. 90
  • Variable-length symbolic compression networks use optimal codewords to assess stochastic, chaotic, epileptic, and time-irreversible dynamics without restricting sequences to equal-degree patterns.Unused-codeword fractions discriminate model systems and EEG seizure data, while minimal cycle bases of ordinal compression networks test time irreversibility.Source: §5.5 · p. 91

6. Real-world applications

This section surveys real-world applications of recurrence, visibility, and transition networks across diverse domains, showing how they extract dynamical information from time series. It covers applications that characterize nonlinear dynamics, detect regime shifts and bifurcations, distinguish system states, and reveal temporal structure and differences across systems.

6.1. Recurrence networks

  • Inter-system recurrence-network asymmetries showed a marked influence of the Indian summer monsoon on the East Asian branch rather than the reverse [191].The result came from cross-transitivities and global cross-clustering coefficients computed for speleothem oxygen-isotope proxies spanning roughly 10,000 years.Source: §6.1 · p. 92
  • Recurrence-network measures distinguished qualitatively different two-phase-flow patterns, with transitivity, assortativity, motifs, and cross-transitivity identifying transitions and increasing dynamical complexity.Cross-transitivity traced transitions between stable stratified and unstable droplet-forming states, while motif distributions revealed increasing heterogeneity whose ranking remained conserved across experimental conditions.Source: §6.1 · pp. 92–93
  • Recurrence-network applications also revealed spatial vegetation-complexity patterns, increasing electrochemical complexity without the speculated phase transition, and discrimination of pre-eclampsia and epileptic states.The vegetation analysis used 14-year global remote-sensing records, while medical applications used heartbeat, blood-pressure, and EEG signals.Source: §6.1 · pp. 92–93
  • Recurrence-network transitivity and average path length identified three statistically significant African climate-transition epochs at 3.5–3.0, 2.25–1.6, and 1.1–0.7 Ma B.P..The epochs were detected in at least two records and associated with distinct climatic mechanisms, although the Middle Pleistocene transition was not detected at ODP site 659, possibly because of local sensitivity or data limitations.Source: §6.1 · pp. 94–95, Figure 30 · p. 94

6.2. Visibility graphs

  • Visibility-graph methods have been applied broadly to turbulence, finance, physiological and EEG signals, hydrology, weather, seismicity, catastrophes, and multiparticle-production data [62] [273] [314] [410].Applications used motifs, topological measures, fractal properties, time-reversal asymmetry, and modified visibility algorithms to characterize system dynamics or detect transitions.Source: §6.2 · pp. 95–96
  • Horizontal-visibility-graph tests found robust time-irreversibility signatures during the Medieval Climate Anomaly–Little Ice Age transition in both North Atlantic sediment records, supporting a nonlinear oceanic regime shift.Clustering-based tests gave pC < 0.05 for Rapid 21-COM and pC ≤ 0.1 for CR 948/2011, whereas degree-based tests had higher p-values of about 0.2 but matching timing.Source: §6.2 · pp. 96–97, Figure 31 · p. 97
  • Negatively inverting sunspot series exposed a bimodal visibility-degree distribution with extreme degrees at least two orders of magnitude above most vertices, while power-law hypotheses were rejected.The original series instead showed heavy-tailed degree distributions; the inverted construction specifically captured local minima that standard visibility graphs overlook.Source: §6.2 · p. 98, Figure 32 · p. 98
  • Sunspot visibility graphs showed broadly distributed strong maxima, clustered strong minima, and communities corresponding mainly to consecutive approximately 11-year solar cycles.Stronger but infrequent maxima formed inter-community hub connections because they were visible to more neighboring observations.Source: §6.2 · p. 98
  • Visibility graphs identify solar-cycle hubs with k_i > 15 and show strong positive correlations between degree and betweenness centrality, indicating information-transporting nodes.These network features extend across several consecutive solar cycles and have been suggested as markers of solar cycles.Source: §6.2 · p. 99, Figure 33 · p. 99
  • VG excess degrees reveal hemispheric-predominance transitions around 1925–1935 and 1985–1995, whereas HVG quantities are weaker and show no comparably clear transitions.HVG degrees are typically 2–4 times smaller than VG degrees, while HVG absolute excess degrees are more than one order of magnitude smaller.Source: §6.2 · p. 100, Figure 34 · p. 100
  • The observed long-term VG excess-degree changes are dominated by hemispheric sunspot-area asymmetry, but cannot be fully explained by skewness and HVG-based dynamical changes alone.Complex changes in the sunspot-area probability distribution may affect VG properties beyond fluctuations in skewness.Source: §6.2 · pp. 100–101
  • Temporal changes in graph-based hemispheric predominance lag changes directly associated with total hemispheric sunspot areas, opening a dynamical perspective on North–South asymmetry.Horizontal visibility graphs provide complementary information on hemispheric asymmetries in dynamical properties, while absolute and relative excess degrees show qualitatively similar long-term variability.Source: §6.2 · pp. 99–101

6.3. Transition networks

  • Ordinal transition-network measures successfully track period-doubling bifurcations, chaos, and the diode resonator’s interior crisis, with network changes linked to attractor filling.Mean out-degree, mean shortest path length, and diameter respond to dynamical changes, although they support relative discrimination and change-point detection rather than absolute periodic-versus-chaotic classification.Source: §6.3 · pp. 101, 103
  • Ordinal-network transitional complexity clearly discriminates ECG recordings of normal sinus rhythm, ventricular tachycardia, and ventricular fibrillation.A two-dimensional measure based on global node out-link entropy at short and long embedding lags captures multiscale complexity, with the 81-record dataset showing clear pathological-group separation.Source: §6.3 · p. 103, Figure 36 · p. 104
  • Global node out-link transitional complexity distinguishes age-related ECG dynamics more effectively than standard permutation entropy.SGNE is generally higher for elderly subjects on short time scales, lower on long time scales, and more variable than in young subjects.Source: §6.3 · p. 103

7. Software implementation – pyunicorn

pyunicorn integrates complex-network and nonlinear-time-series methods in a performant, modular, flexible, and open-source Python package applicable across diverse research domains. Its five subpackages support general, functional, climate, time-series, and utility workflows, while related tools provide narrower recurrence-network, visualization, or method-specific capabilities.

Scope and purpose

  • pyunicorn unites complex network theory and nonlinear time series analysis in a performant, modular, flexible Python package applicable to climatology, neuroscience, hydrology, economics, and finance [73].The package is presented as broadly applicable to large time-series analyses beyond its climatology tutorial examples.Source: §7 · p. 104

Open-source infrastructure

  • The code is fully open sourced under the BSD 3-Clause license and has evolved through user contributions as shared infrastructure for scientific data analysis.The package is intended as an integrated container for conceptually related methods developed by participating research groups.Source: §7 · p. 104

Software structure

  • Its five subpackages cover general and spatial or multiplex networks, functional networks, climate networks, recurrence and visibility graphs, surrogate generation, MPI parallelization, and interactive navigation.The timeseries subpackage covers the reviewed recurrence-network and horizontal-visibility-graph methods, except ordinal pattern transition networks.Source: §7 · p. 104

Computational design

  • Compiled implementations on sparse data structures support large networks and time series, while object-oriented modularity combines methods such as complex networks and recurrence plots into recurrence-network analysis.This architecture is intended to enable flexible and parsimonious combinations of data structures, methods, and algorithms.Source: §7 · pp. 104–105

Limitations and related tools

  • pyunicorn may facilitate methodological development, but users should ensure extensions are theoretically well-founded and driven by well-posed, relevant research questions.Other available tools include CRP Toolbox, Mathematica demonstrations, method-specific visibility-graph code, Gephi, and NetworkX.Source: §7 · p. 105

8. Conclusions and future perspectives

The review establishes recurrence networks, visibility graphs, and transition networks as major frameworks for time-series analysis, while identifying future work needed to extend their applicability and demonstrate added value beyond standard methods.

8.1. Conclusions

  • Recurrence networks, visibility graphs, and transition networks transform time series into network representations that characterize dynamics, distinguish regimes, detect transitions, test reversibility, and support prediction.These approaches also broaden analysis to short, complex, and multivariate data while providing measures of phase-space geometry.Source: §8.1 · p. 105

8.2. Future perspectives

  • Future research should establish threshold-invariant recurrence-network findings and demonstrate network methods’ added value through applications that reveal features missed by standard time-series approaches.For recurrence networks, the threshold ε controls link density, and network characteristics become trivial under full connectivity [17].Source: §8.2 · pp. 105–106

8.2.1. Evolving and temporal network analysis for time series

  • Evolving and temporal networks could represent nonstationary dynamics, but sliding-window analyses require objective choices of window length and overlap and may miss temporal ordering or transient links.Temporal formulations can encode edge lifetimes, while short windows improve resolution but obscure slow trends and long windows lose event-level information.Source: §8.2.1 · p. 106

8.2.2. Multilayer and multiplex network analysis for multiple time scale time series

  • Multilayer and multiplex networks should explicitly represent multiple time scales, using scale-sensitive preprocessing such as empirical mode decomposition or wavelet transforms.This is important when higher-frequency variability obscures larger-scale trends, as in climate and neuroscience data.Source: §8.2.2 · pp. 106–107

8.2.3. The inverse problem of time series regeneration from networks

  • Recovering time series from network representations remains challenging because topology does not preserve vertex order, although reconstruction algorithms have recovered attractor topology under specific conditions.Future work must evaluate algorithmic parameters, general performance, applicability, and null hypotheses for surrogate analysis.Source: §8.2.3 · p. 107

8.2.4. Combining data mining tools with time series network approaches

  • Combining network approaches with data mining, statistics, machine learning, and visualization is an important largely unexplored direction for extracting knowledge from increasingly complex time-series datasets.Such integration could address high dimensionality, feature correlations, and noise while providing novel insights into complex systems.Source: §8.2.4 · pp. 107–108

8.2.5. Building network models for time series prediction

  • Network methods have primarily diagnosed observed dynamics rather than forecast them, leaving network-based time-series modeling and prediction as an open research avenue.Existing nonlinear-dynamics approaches commonly use local state-space models to predict future trajectory points [17].Source: §8.2.5 · p. 108

Appendix A. Mathematical models

The appendix lists the mathematical models used in the paper’s examples, including canonical chaotic systems, an auto-regressive process, and diffusively coupled Rössler systems. It also specifies key parameter settings and noise or coupling assumptions where provided.

Mathematical models

  • The examples use the Lorenz system with r = 28, σ = 10, and β = 8/3.These parameters define the Lorenz-model configuration used in the examples.Source: Appendix A · p. 109
  • The model set also includes the Rössler system, the Hénon map, and an auto-regressive process of order p.For the auto-regressive process, ϕj are real-valued coefficients and εt is Gaussian white noise with zero mean and unit variance.Source: Appendix A · p. 109

Coupled systems

  • The appendix includes two Rössler systems diffusively coupled through the second y-component, with symmetric coupling when µ12 = µ21.ν denotes frequency mismatch, while µ12 and µ21 denote coupling strengths.Source: Appendix A · p. 109
  • It also includes three diffusively coupled Rössler systems coupled through the x component, with µ as the coupling strength.The systems are indexed by k = 1, 2, 3.Source: Appendix A · p. 109

Model parameters

  • The appendix further identifies κ as the single control parameter of one listed system.The supplied passage does not specify the associated model equation.Source: Appendix A · p. 109

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