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Recurrence-based time series analysis by means of complex network methods
Reik V. Donner, Michael Small, Jonathan F. Donges, Norbert Marwan, Yong Zou, Ruoxi Xiang, Jürgen Kurths
TL;DR
Time-series analysis needs ways to characterize higher-order dynamical structure beyond conventional approaches, especially through recurrence in phase space. This paper reviews complex-network methods, emphasizing recurrence-plot-based constructions and discussing their use across paradigmatic and real-world data. The reviewed measures capture structural features complementary to other linear and nonlinear time-series methods, while their interpretation is limited by method-specific assumptions and information loss.
Problem
The paper addresses how complex-network concepts can analyze dynamically relevant higher-order properties of time series through recurrence-related representations.
Method
The paper reviews network constructions from time series, with emphasis on recurrence plots, recurrence networks, and their technical issues and applications.
Results
Complex-network measures provide information about phase-space structural features that is complementary to other time-series analysis methods.
Takeaways & Limitations
Recurrence-based complex networks enrich the knowledge obtained from existing linear and nonlinear approaches to time-series analysis.
Takeaways & Limitations
Recurrence-based networks do not account for temporal correlations, and removing short-lag recurrence points can also remove true recurrences.
Abstract
from arXiv · showhide
Complex networks are an important paradigm of modern complex systems sciences which allows quantitatively assessing the structural properties of systems composed of different interacting entities. During the last years, intensive efforts have been spent on applying network-based concepts also for the analysis of dynamically relevant higher-order statistical properties of time series. Notably, many corresponding approaches are closely related with the concept of recurrence in phase space. In this paper, we review recent methodological advances in time series analysis based on complex networks, with a special emphasis on methods founded on recurrence plots. The potentials and limitations of the individual methods are discussed and illustrated for paradigmatic examples of dynamical systems as well as for real-world time series. Complex network measures are shown to provide information about structural features of dynamical systems that are complementary to those characterized by other methods of time series analysis and, hence, substantially enrich the knowledge gathered from other existing (linear as well as nonlinear) approaches.
1. Introduction
Recurrence-based methods represent repeated visits in phase space and use recurrence plots to characterize dynamical behavior. Reinterpreting recurrence matrices as network adjacency matrices extends nonlinear time-series analysis with structural measures complementary to established approaches.
- Recurrence concepts: Recurrence describes when a phase-space state revisits a sufficiently similar neighborhood, a concept applicable to reconstructed or fully specified phase spaces.The definition uses sampled states and allows generalization to unequally sampled data despite assuming constant sampling time in its basic formulation.
- Recurrence concepts: Recurrence plots encode recurrences as a binary matrix, with alternative constructions based on nearest neighbors or a fixed phase-space threshold ε.Nearest-neighbor plots preserve a fixed local recurrence rate, whereas threshold-based plots mark states closer than ε under a chosen norm.
- Recurrence plots: Recurrence-plot structures distinguish dynamical regimes: periodic motion yields long uninterrupted diagonals, chaos shorter diagonals, and uncorrelated noise mainly isolated points.These structural features provide small-scale and large-scale descriptors of the underlying dynamics.
- Recurrence-based analysis: Recurrence quantification analysis measures individual points and diagonal or vertical lines, while recurrence networks provide another structural representation of the same recurrence information.The reviewed recurrence-based measures have also been used to estimate dynamical invariants and study couplings and synchronization.
- Recurrence networks: Mapping the recurrence matrix R to an unweighted network adjacency matrix A creates a nonlinear time-series method that connects phase-space structure with complex-network topology.The approach is illustrated using the Lorenz system and its recurrence network.
2. Transforming time series into complex networks
Time-series network methods are organized by how observations or trajectory segments determine vertices and edges. Most are recurrence-related, but their treatment of locality, geometry, temporal order, and causality differs.
- Method classes: The methods fall into proximity networks, visibility graphs, and transition networks, based respectively on mutual proximity, convexity, or transition probabilities.This classification summarizes the main ways observational time series are transformed into network representations.
- Method classes: Except for visibility graphs, the approaches are related to recurrence, with proximity networks using data-adaptive local neighborhoods and transition networks using fixed phase-space classes.Proximity neighborhoods vary by vertex, whereas transition-network classes arise from rigid coarse-graining.
- Structural relationships: Recurrence, proximity, and transition representations share a binary-matrix foundation that supports statistical analysis of network topology.Proximity and transition networks, as well as threshold-based recurrence plots, can be generalized beyond a single threshold.
- Proximity networks: Proximity networks include cycle, correlation, and recurrence networks, whose topological measures are invariant under relabeling vertices.This permutation invariance distinguishes them from traditional time-series methods that explicitly retain observation order.
- Interpretation: Recurrence networks are spatial networks embedded in phase space, so graph paths differ from causal trajectories; transition networks preserve causal relationships.Distances are defined using standard metrics such as Euclidean or Manhattan distance.
2.1. Cycle networks
Cycle networks represent individual oscillatory cycles as vertices and connect segments that behave similarly. They avoid explicit time-delay embedding and can distinguish dynamical regimes through network structure.
- Construction: Cycle networks identify individual cycles in pseudo-periodic time series as vertices and connect pairs whose trajectory segments behave similarly.The method was proposed for systems with pronounced oscillations, including Lorenz and Rössler examples.
- Construction: Cycle similarity can be measured by the maximum cross-correlation obtained by sliding the shorter cycle relative to the longer one.The compared cycles may have different lengths, and the correlation index maximizes over admissible shifts.
- Properties: Cycle networks avoid explicit time-delay embedding and are more robust to additive noise when the noise magnitude remains small enough to preserve cycle identification.The method still requires clear identification of individual cycles from the time series.
- Properties: Because cycle networks are invariant under reordering cycles, their structure reflects orderly variation among cycles rather than their original temporal sequence.Linear and periodic systems tend to produce random-looking cycle networks, whereas chaotic and nonlinear systems generate highly structured ones.
- Applications: Vertex, edge, and mesoscale properties such as clustering can distinguish dynamical-system classes and help locate unstable periodic orbits.The cited discussion links orderly cycle variation to differences between chaotic or nonlinear dynamics and linear or noise-driven systems.
2.2. Correlation networks
Correlation networks connect embedded time-series states when their Pearson correlation exceeds a threshold, but this similarity can conflate shifted trajectories with proximity in phase space. For the Lorenz example, the resulting network reveals community structure that differs from the attractor’s two-scroll geometry.
- Construction: Correlation networks represent embedded state vectors as vertices and connect pairs whose Pearson correlation exceeds a threshold r.This criterion is equivalent to a recurrence condition using 1 − r as a proximity measure, subject to sufficiently large embedding dimension.
- Limitations: Embedding and correlation estimation can lose short-term dynamical information and introduce spurious correlations.The method usually requires a sufficiently large embedding dimension, while embedding itself may induce correlations.
- Lorenz example: Correlation connects Lorenz intervals t = 0...7 and t = 24...28 despite their occupying different parts of the attractor.Their internal dynamics appear similar, but one interval is shifted in the x- and y-coordinates.
- Limitations: The Lorenz comparison demonstrates that correlation-based similarity must be distinguished from true metric distance.Mean-position removal in correlation estimation makes shifted trajectory segments appear similar under the correlation criterion.
- Lorenz example: The correlation network forms two ring-like communities that do not correspond to the Lorenz attractor’s two scrolls.The observed grouping is reasonably associated with the orientation of arc-like embedding vectors around the centers.
2.3. Recurrence networks
Recurrence networks transform recurrence matrices into graphs whose topology characterizes phase-space structure without explicitly preserving temporal ordering. Different constructions emphasize different geometric properties, while network measures complement RQA and other time-series methods.
- Network definition: A recurrence network uses a time series’ recurrence matrix as its adjacency matrix, after removing the line of identity.This construction links recurrence-based phase-space analysis to complex-network measures.
- Network definition: Unlike RQA, recurrence-network topology omits temporal ordering and therefore primarily reflects spatial, dynamically invariant properties of the phase-space system.Network analysis provides a distinct characterization based mainly on geometric relationships among states.
- k-nearest neighbor networks: In k-nearest neighbor networks, each observation links to its k closest phase-space neighbors, producing generally directed graphs with fixed out-degree and mean in-degree k.The in-degree pattern indicates local attractor density: low in-degree marks relatively sparse regions, whereas high in-degree marks dense regions.
- Adaptive nearest neighbor networks: Adaptive nearest neighbor networks enforce exactly E0 undirected edges per vertex, providing precise edge control and minimum degree E0.Their motif distributions distinguish periodic, chaotic, and hyper-chaotic dynamics, with non-transitive motifs more common in chaotic systems.
- ε-recurrence networks: Nearest-neighbor networks are invariant under monotonic rescaling but lack a direct relationship between network properties and invariant density, while ε-recurrence networks restore that relationship.ε-recurrence networks define neighborhoods by fixed phase-space distance and can relate local properties directly to phase-space structure.
- Network analysis and RQA: Global measures such as clustering, transitivity, and path length trace qualitative dynamical changes, but their interpretation differs between discrete maps and continuous systems.Network measures capture complementary information to RQA and other established time-series methods, so combining approaches can provide additional information.
2.4. Other approaches
The paper surveys visibility and transition networks as alternative ways to represent time series, highlighting their structural interpretations and important limitations. Visibility graphs reveal temporal organization through hubs and communities, while transition networks encode coarse-grained phase-space transitions.
- Visibility graphs: Visibility graphs connect intervisible observations and have been applied to time series, especially stochastic processes with fractal structure.Their degree distribution can be related to the Hurst parameter, but broader links to phase-space properties remain limited.
- Visibility graphs: Visibility graphs typically contain hubs at local maxima, producing communities whose clusters reflect the temporal order of observations.These features often yield scale-free degree distributions associated with fractal properties of the time series.
- Transition networks: Transition networks coarse-grain time-series values into classes and represent class-to-class transition probabilities as weighted directed edges.This is equivalent to symbolic discretization with stationary transition probabilities.
- Transition networks: Transition networks identify phase-space regions important for causal evolution through betweenness centrality and related measures.They lose information about small-amplitude variations and depend on the full class definition, although coarse-graining can help extract dynamics from noisy data.
- Transition networks: For the Lorenz system, transition-network connectivity reveals the attractor’s spatial structure through near-diagonal links and higher connectivity within its two scrolls.The topology reflects closely neighboring successive observations under dense sampling, with some transitions bridging multiple coarse-grained cells.
3. Practical considerations
Practical use of recurrence-based networks requires careful choices of cycle segmentation, sampling, embedding, recurrence threshold, and network construction. These choices affect robustness, comparability, and quantitative network measures, although some adaptive-neighbor variations diminish for larger datasets.
- Cycle networks: Cycle networks require dividing the time series into distinct cycles and comparing their mutual proximity with application-dependent measures.Cycles are preferably defined at peaks or troughs, and sufficiently high sampling is needed so cycle lengths remain reasonably large.
- Recurrence networks: Sojourn points from strong tangential motion can create artificial recurrence-network links between temporally adjacent observations in a small phase-space region.Removing them requires temporal-separation rules or direct matrix filtering, but these choices can discard true recurrences or introduce an additional parameter.
- Sampling and embedding: Longer time series improve nearest-neighbor spatial resolution, while ε-recurrence networks use the threshold ε to control spatial connectivity.Sampling should resolve the attractor’s covered phase space, and embedding parameters should be selected appropriately.
- Adaptive nearest-neighbor networks: Adaptive nearest-neighbor networks depend slightly on the processing order, but this variation does not produce significant structural changes for moderate to large N.For the studied Rössler and Lorenz systems, the mean Hamming distance from permutations suggests µ_H ∼ N^-1.
- Adaptive nearest-neighbor networks: Motif-based classification is reasonably robust to the number of edges per node, whereas 4-motifs are a practical compromise because lower orders are uninformative and higher orders become intractable.The limitation is computational rather than a demonstrated loss of dynamical information.
- ε-recurrence networks: Threshold-selection heuristics can be misleading because general criteria depend crucially on sampling and embedding.The turning-point criterion assumes a unique turning point in the RR(ε) relationship, an assumption that may not hold reliably.
- ε-recurrence networks: The average path length L is approximately inversely proportional to ε, while transitivity and clustering approach one for sufficiently large ε.Assortativity varies more diversely with ε and approaches one when ε is large.
- ε-recurrence networks: Fixing recurrence rate RR rather than ε gives ε-recurrence networks with approximately equal edge counts, facilitating topological comparisons across time series.This choice is especially important when comparing systems with different dynamical properties, such as periodic and chaotic orbits.
4. Applications
Applications show recurrence-based networks can classify dynamical regimes, identify transitions in real-world records, and reveal structure in experimental signals. These analyses provide results that complement classical time-series methods while relying on assumptions suited to irregularly sampled data.
- 4.1. Classification of dynamical systems: The clarinet application found 17 of 20 distinct tones in one motif superfamily, with E3, F4, and B6 forming a closely related second family.The three tones were described as possible less-stationary outliers, but the explanation was preliminary.
- 4.1. Classification of dynamical systems: Motif prevalence classified clarinet tones as most similar to chaotic or hyperchaotic dynamics rather than noisy periodic behavior.Seventeen of 20 tones shared one motif superfamily, while three belonged to a closely related family.
- 4.2. Identification of dynamical transitions: Recurrence-network measures detected variability and distinct epochs in average path length, transitivity, assortativity, and network diameter in the paleoclimate record.Average path length increased around 3.5–3.3 Ma, 2.1 Ma, and 1.9–1.7 Ma BP, while transitivity increased across additional intervals.
- 4.2. Identification of dynamical transitions: The identified paleoclimate intervals were robust and correlated with major climate-system transitions, including the end of the Pliocene optimum.These intervals had not been found using spectral analysis or breakpoint regression.
5. Summary
The paper constructs complex networks from dynamical recurrences by treating recurrence information as network connectivity. It concludes that these networks offer a complementary approach for characterizing dynamics, detecting transitions, and identifying invariant structures, while depending on phase-space geometric and sampling assumptions.
- 5. Summary: Recurrence-based networks exploit the duality between recurrence matrices and network adjacency matrices to analyze time series.The approach combines recurrence plots with complex-network methods.
- 5. Summary: Complex-network measures can characterize and classify dynamics, detect dynamical transitions, and identify invariant substructures.These capabilities are presented as applications of recurrence-based networks.
- 5. Summary: Recurrence-based complex networks provide a promising and complementary view for studying dynamical systems.The paper presents this potential through initial applications across dynamical and real-world systems.
- 5. Summary: Recurrence-network characteristics have geometric interpretations in the underlying phase space and do not explicitly require equally spaced observations.The method instead assumes that observations represent the underlying phase-space distribution in a statistically reasonable way.