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Unveiling the higher-order organization of multivariate time series

Andrea Santoro, Federico Battiston, Giovanni Petri, Enrico Amico

arXiv:2203.10702v2physics.soc-phcs.SIq-bio.NC

TL;DR

Direct higher-order network measurements are often inaccessible, despite the relevance of group interactions in complex systems. This paper uses topological markers to reconstruct higher-order dynamics from multivariate time series, distinguishing coupled-map regimes and revealing structure in brain, financial, and epidemic data.

  • Problem

    Direct higher-order network measurements are often inaccessible, while many complex-system activity patterns are recorded at the level of nodes rather than groups.

    Source: DISCUSSION · p. 9

  • Method

    The paper introduces a topological approach that dynamically reconstructs higher-order interactions from multivariate time series.

    Source: DISCUSSION · p. 9

  • Results

    The higher-order indicators robustly separate phases in coupled chaotic maps and reveal brain, financial, and epidemic patterns, including complementary insights not trivially deducible from edge-wise approaches.

    Source: DISCUSSION · pp. 9–10

  • Takeaways & Limitations

    Higher-order markers provide complementary information about dynamical group dependencies in the examined real-world time series.

    Source: DISCUSSION · pp. 9–10

    The main analysis focuses on interactions up to three-body structures and H1 cycles; without four-body interactions, 3D cavities have infinite persistence and add no meaningful information at that order.

    Source: S2.2. Impact of 3D-cavities · p. 21

Abstract

from arXiv · show

Time series analysis has proven to be a powerful method to characterize several phenomena in biology, neuroscience and economics, and to understand some of their underlying dynamical features. Despite a plethora of methods have been proposed for the analysis of multivariate time series, most of them neglect the effect of non-pairwise interactions on the emerging dynamics. Here, we propose a novel framework to characterize the temporal evolution of higher-order dependencies within multivariate time series. Using network analysis and topology, we show that, unlike traditional tools based on pairwise statistics, our framework robustly differentiates various spatiotemporal regimes of coupled chaotic maps, including chaotic dynamical phases and various types of synchronization. Hence, using the higher-order co-fluctuation patterns in simulated dynamical processes as a guide, we highlight and quantify signatures of higher-order patterns in data from brain functional activity, financial markets, and epidemics. Overall, our approach sheds new light on the higher-order organization of multivariate time series, allowing a better characterization of dynamical group dependencies inherent to real-world data.

RESULTS

The section validates higher-order indicators on coupled chaotic maps, then applies global and local measures to brain, financial, and epidemic time series. The indicators distinguish dynamical regimes and reveal distinct real-world patterns at global and local scales.

RESULTS

  • The framework z-scores nodal series, forms and rescales group-product co-fluctuations, assigns concordant groups positive and discordant groups negative signs, then analyzes weighted simplicial complexes and their filtrations.The reported analysis focuses on interactions up to triangles; violating triangles encode higher-order patterns, while persistent H1 cycles yield the hyper-complexity measure.Source: RESULTS · pp. 2–4

Global and local topological markers classify different dynamical regimes

  • Hyper coherence is higher in synchronized CML regimes and lower in chaotic or turbulent ones; hyper complexity is lowest for periodic patterns and highest for chaotic states.The violating-triangle edge-weight distribution shifts from well-peaked for periodic series toward fat-tailed for fully developed turbulence, while homological-scaffold distributions also change from periodic to chaotic regimes.Source: RESULTS · pp. 4–5, Figure 2 · p. 5
  • Hyper coherence and hyper complexity achieve ICCs of approximately 0.95 and 0.96, versus approximately 0.57 for RSS and approximately 0 for Pearson correlation across five CML regimes.Static s-information reaches ICC ≈0.98, but typically uses temporal windows; the topological framework provides instantaneous information.Source: RESULTS · pp. 5–6, Figure 3 · p. 6

Real-world complex systems exhibit non-trivial hyper coherence structure

  • Real-world hyper-coherence distributions differ significantly from independently reshuffled nulls (all p-values < 10^-10) and show distinct profiles across brain, financial, and epidemic data.At rest, brain data are mostly associated with chaotic and some partially synchronized states; financial distributions are bimodal, and epidemic distributions vary by disease.Source: RESULTS · pp. 6, 8, Figure 4 · p. 7

Hyper complexity decomposition provides detailed information about dynamical regimes

  • Hyper-complexity contributions separate the five CML regimes: partially synchronized signals mix Full Coherence and Coherence Transition, while chaotic signals are mainly Full Coherence and Full Decoherence.At rest, most brain states fall between chaotic and partially synchronized regimes; the three contributions sum to total hyper complexity and provide information different from hyper coherence.Source: RESULTS · pp. 7–8

Real-world applications of higher-order topological markers

  • In the most coherent resting-brain frames, violating-triangle projections emphasize sensorimotor areas; low-hyper-complexity frames instead emphasize the Default Mode Network.The latter frames are associated with a more synchronized dynamical phase.Source: RESULTS · pp. 8–9
  • Violating-triangle nodal strength captures the onset of financial instability in 2002 and 2007–2008, while homological-scaffold strength identifies consumer goods, basic materials, and oil and gas during the 2007 recession.The instability periods show increased stock-price synchronization, unlike the more stable intervals described in the passage.Source: RESULTS · p. 9
  • Higher-order measures classify US infectious diseases with around 85% SVM accuracy under 10-fold cross-validation repeated 50 times, and the embedding shows disease and state-specific outbreak patterns.Gonorrhea and chlamydia largely overlap in the embedding.Source: RESULTS · p. 9

DISCUSSION

The higher-order framework separates coupled chaotic-map regimes and reveals distinct patterns in brain, financial, and epidemic time series. Its framewise estimates provide an alternative when direct higher-order measurements are inaccessible.

DISCUSSION

  • The framework reconstructs higher-order dynamics from multivariate time series and provides framewise estimates and instantaneous topological snapshots of system configurations.The authors position it as an alternative where direct higher-order network measurements are often inaccessible.Source: DISCUSSION · pp. 9–10
  • Higher-order indicators robustly separate several coupled-map dynamical phases; local triangle-weight distributions are well-peaked for synchronized periodic series and fat-tailed for chaotic series.The authors report that these phases appear distinguishable only with higher-order statistics.Source: DISCUSSION · p. 9
  • At rest, brain higher-order dynamics mainly oscillates between fully developed turbulence and partial synchronization, while sensorimotor and default-mode regions show complementary higher-order patterns.Maximally coherent structures reflect sensorimotor areas, while low hyper-complexity points coarsely capture the default-mode network.Source: DISCUSSION · pp. 9–10
  • Higher-order structure magnitude discriminates financial crises from stability, while low hyper-complexity points capture changing sector roles and industry heterogeneity.The authors suggest these markers may help identify the building up of systemic risk.Source: DISCUSSION · p. 10
  • For US epidemic outbreaks, temporal higher-order measures classify diseases to a great extent, and planar embeddings reveal clusters with distinct spatiotemporal patterns.The authors note a limitation as forecast length increases.Source: DISCUSSION · p. 10

METHODS

The framework turns instantaneous co-fluctuations into weighted simplicial complexes, then uses filtration and persistent homology to characterize their higher-order structure. It also derives network-level summaries and analyzes three real-world datasets.

METHODS

  • The method generalizes edge-centric co-fluctuations to groups of time series, z-scores the group-product series across time, and assigns signs to distinguish concordant from discordant interactions.The reported implementation considers interactions up to k = 2, so triangles are the higher-order structures.Source: METHODS · pp. 10–11
  • At each time point, the co-fluctuations form a weighted simplicial complex in which every simplex's subfaces must also be present.The complex is built from edges and triangles in this implementation.Source: METHODS · p. 11
  • A descending-weight filtration tracks 1D cycles in H1, while triangles entering before their edges are recorded as violations and used to define hyper coherence.Hyper coherence is the fraction of violating coherent triangles among all coherent triangles; the average number of already-present edges per violating triangle is also measured.Source: METHODS · p. 11
  • The hyper complexity indicator is the Wasserstein distance between the H1 persistence diagram and the empty diagram.The homological scaffold weights each edge by the summed persistence of cycles containing it, while downward projections summarize triangle violations at edge and node levels.Source: METHODS · pp. 11–12
  • The real-world analyses use resting-state fMRI, NYSE stock prices, and historical US infectious-disease data.The fMRI data comprise 100 unrelated HCP subjects; the supplied passage also describes preprocessing steps for this dataset.Source: METHODS · p. 12

S1. FILTRATION AND PERSISTENT HOMOLOGY

The analysis orders co-fluctuation simplices into a filtration and tracks the birth and death of its 1D cycles. Hyper complexity is decomposed by whether those cycles involve coherent or decoherent structures.

S1. FILTRATION AND PERSISTENT HOMOLOGY

  • The filtration orders coherent patterns from the empty complex to the full complex, and persistent homology tracks the persistence of 1D cycles in H1.These persistent generators indicate where and when higher-synchronised regions emerge.Source: S1. FILTRATION AND PERSISTENT HOMOLOGY · p. 18
  • Hyper complexity is the Wasserstein distance from the H1 persistence diagram to the empty diagram, and its contributions separate full coherence, coherence transition, and full decoherence.The three categories distinguish cycles created and closed by coherent structures, cycles formed by coherent and closed by decoherent structures, and cycles created only by decoherent structures.Source: S1. FILTRATION AND PERSISTENT HOMOLOGY · p. 18, Figure S1 · p. 19
  • Barcodes and persistence diagrams encode each topological feature by its birth and death along the filtration; the diagram represents each H1 cycle as a point (wb, wd).Figure S1 presents the three fractional hyper-complexity contributions in a triangular representation.Source: S1. FILTRATION AND PERSISTENT HOMOLOGY · p. 18, Figure S1 · p. 19

S2. GLOBAL BEHAVIOUR OF COUPLED MAP LATTICE (CML)

The coupled map lattice generates five dynamical regimes at different coupling strengths. The space-amplitude plots show their distinct global behaviours.

S2. GLOBAL BEHAVIOUR OF COUPLED MAP LATTICE (CML)

  • N = 119 diffusively coupled fully chaotic maps generate time series for five dynamical states at different coupling strengths.The regimes are Pattern Selection (ε = 0.12), Brownian motion with Defects (ε = 0.08), Spatiotemporal Intermittency II (ε = 0.3), Defect Turbulence (ε = 0.068), and Fully Developed Turbulence (ε = 0.05).Source: Figure S2 · p. 20, S2. GLOBAL BEHAVIOUR OF COUPLED MAP… · p. 20
  • The space-amplitude plot shows the temporal evolution of xi(t) at each lattice site and reflects differences in the five regimes' global behaviour.The accompanying sample time series shows the general behaviour of the regimes at three lattice sites.Source: Figure S2 · p. 20, S2. GLOBAL BEHAVIOUR OF COUPLED MAP… · p. 20
  • The five regimes are presented from ordered to disordered as Pattern Selection, Brownian motion with Defects, Spatiotemporal Intermittency II, Defect Turbulence, and Fully Developed Turbulence.Their reported coupling strengths are respectively ε = 0.12, 0.08, 0.3, 0.068, and 0.05.Source: Figure S2 · p. 20

S2.1. Effect of z-scores and signs assignment in CML time series

Z-scoring and sign assignment shift triangle weights negatively relative to edge weights. This has limited impact on hyper coherence but can bias hyper complexity contributions.

S2.1. Effect of z-scores and signs assignment in CML time series

  • The triangle-weight negative offset can shift hyper complexity from fully coherent toward the other two contributions, without restructuring the whole simplicial filtration.The passage characterizes this as a biasing effect.Source: S2.1. Effect of z-scores and signs… · p. 21, Figure S3 · p. 21
  • The negative offset has no deep impact on hyper coherence, which is defined from violating coherent triangles relative to all coherent triangles.Both sets are restricted to triangles with positive weight.Source: S2.1. Effect of z-scores and signs… · pp. 20–21
  • Without sign remapping, a triangle's weight cannot distinguish synchronous co-activation from decoherent triangles that have the same weight.Figure S3 compares edge and triangle weight distributions with and without sign assignment across the five CML states; error bars are standard deviations over 240 time points.Source: S2.1. Effect of z-scores and signs… · p. 20, Figure S3 · p. 21

S2.2. Impact of 3D-cavities

The analysis extends to 4-body interactions to examine H2 generators, or 3D-cavities. Their persistence diagrams vary across the five CML regimes and may help distinguish them alongside H1.

S2.2. Impact of 3D-cavities

  • The analysis includes interactions up to 4-body to investigate 3D-cavities, the generators of homology group H2.The main-text analysis focused on H1 and interactions up to 3-body.Source: S2.2. Impact of 3D-cavities · pp. 21–22
  • Without 4-body interactions to close them, 3D-cavities have infinite persistence because they are born but do not die in the filtration.Source: S2.2. Impact of 3D-cavities · p. 21
  • H2 persistence diagrams differ across the five CML regimes and may help further differentiate them when considered with H1.The diagrams use 10 realizations and 240 time points per block for CMLs with N = 50.Source: S2.2. Impact of 3D-cavities · p. 22, Figure S4 · p. 22

S2.3. Higher-order approach on Gaussian multivariate time series

The analysis compares resting-state fMRI hyper coherence with independently shuffled fMRI and Gaussian multivariate time series. The shuffled-fMRI distribution is reported as statistically equivalent to the Gaussian-series distribution.

S2.3. Higher-order approach on Gaussian multivariate time series

S3. COMPARISON OF TIME SERIES APPROACHES TO DISTINGUISH THE CML REGIMES

Higher-order measures distinguish the five CML regimes more effectively than RSS and pairwise statistical approaches in the reported comparisons. Across the five regimes, their ICC values exceed RSS's, and static pairwise measures fail to differentiate the regimes.

S3. COMPARISON OF TIME SERIES APPROACHES TO DISTINGUISH THE CML REGIMES

  • RSS is the root sum of squared edge-time-series co-fluctuations and serves as a proxy for their collective amplitude.The edge time series is formed from products of the original time-series z-scores.Source: S3. COMPARISON OF TIME SERIES… · p. 24
  • The comparison includes information-theoretic higher-order measures, a dyadic network and random-matrix approach, and Pearson correlation.Source: S3. COMPARISON OF TIME SERIES… · p. 24
  • ICC was ≈0.96 for hyper complexity, ≈0.95 for hyper coherence, ≈0.91 for average edge violation, and ≈0.57 for RSS across the five CML regimes.The values are averaged over 100 independent realizations.Source: S3. COMPARISON OF TIME SERIES… · p. 24, Figure S6 · p. 24
  • Static measures were computed separately within each 240-time-point regime interval because they require a time window rather than a single frame.Results were aggregated across 100 independent realizations.Source: S3. COMPARISON OF TIME SERIES… · p. 25
  • Static higher-order measures differentiated the five CML regimes, whereas Pearson correlation and filtered ρ based on random matrix theory did not.The comparison also reports the same conclusion using Jensen-Shannon divergence as with ICC.Source: S3. COMPARISON OF TIME SERIES… · pp. 25–26, Figure S7 · p. 25

S4. IMPACT OF DIFFERENT NULL MODELS

The block and phase-randomization null models test whether higher-order indicators retain regime structure under alternative surrogates. The block null retains some regime-specific co-fluctuation features but loses the hyper-coherence order-disorder ranking, while phase randomization yields flat distributions for both indicators.

S4. IMPACT OF DIFFERENT NULL MODELS

S4.2. Block null model in financial time series

Block-shuffled financial null models retain a hyper-coherence distribution distinct from the empirical data. Across real-world systems, edge-based measures provide only partial information: their relationship to higher-order patterns varies by system, and RSS performs poorly for distinguishing infectious diseases.

S4.2. Block null model in financial time series

  • Even the 9-block null differs from empirical financial hyper coherence: its distribution is almost 4-modal versus the empirical bimodal distribution (p < 10^-10).The finer block nulls increasingly approximate the empirical distribution, but the 9-block model remains statistically distinct.Source: S4.2. Block null model in financial… · p. 30, Figure S10 · p. 30

S5. EDGE-BASED INDICATORS IN REAL-WORLD SYSTEMS

  • RSS distributions differ from shuffled nulls for brain and financial data but are similar for epidemic data, and do not reveal brain or financial temporal regimes.The Gonorrhea comparison reports p < 0.001 with the Kolmogorov-Smirnov test.Source: S5. EDGE-BASED INDICATORS IN… · p. 31, Figure S11 · p. 31
  • Projecting higher-order measures onto nodes can discard moderate-to-high amounts of information and may therefore mislead if not carefully interpreted.The local comparison maps violating triangles, homological scaffold, and edge time series onto nodes.Source: S5. EDGE-BASED INDICATORS IN… · pp. 31–32, Table S1 · p. 33
  • In brain data, top hyper-coherent patterns resemble top- and bottom-RSS patterns, while other higher-order patterns are not fully captured by edge measures; in finance, top hyper-coherent patterns have ρ ≈0 with edge patterns.The brain comparison is also reported in the qualitative map analysis; the financial near-zero correlation indicates dissimilar patterns.Source: S5. EDGE-BASED INDICATORS IN… · pp. 31, 33, Figure S13 · p. 34
  • RSS temporal evolution does not effectively classify US infectious diseases, yielding around 67% accuracy with repeated 10-fold cross-validation.The evaluation repeats 10-fold cross-validation 50 times.Source: S5. EDGE-BASED INDICATORS IN… · p. 33

S6. EFFECT OF NOISY FLUCTUATIONS IN BRAIN DATA

The authors tested whether head-motion noise alters higher-order brain maps. Despite a single-frame noise caveat, maps with and without high-motion volumes were highly correlated.

S6. EFFECT OF NOISY FLUCTUATIONS IN BRAIN DATA

S7.1. Role of thresholds in brain maps

The supplementary analyses find consistent brain maps across peak-selection thresholds and distinct functional-network patterns for two higher-order indicators. They also describe a disease-classification comparison using higher-order features and several classifiers.

S7.1. Role of thresholds in brain maps

S7.2. Higher-order indicators in the brain’s functional networks

S7.3. Comparison of classifiers for the US historical data

  • The US state-level infectious-disease classifier comparison evaluates average accuracy and weighted F1 using hyper coherence, average edge violations, and three hyper-complexity features.The compared classifiers are Gaussian NB, RBF SVM, Decision Tree, Random Forest, and k-NN with k=5.Source: Table S2 · p. 38

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