[Paper Review] Unveiling the higher-order organization of multivariate time series

Andrea Santoro, Federico Battiston+2 more|arXiv (Cornell University)|Mar 21, 2022

Complex Systems and Time Series Analysis4 citations

TL;DR

This paper introduces a novel topological framework using higher-order network structures—specifically hypergraphs and homological scaffolds—to uncover group-level dependencies in multivariate time series. By analyzing co-fluctuation patterns and persistent topological features, the method reveals distinct dynamical regimes in chaotic systems and identifies robust signatures of higher-order organization in brain, financial, and epidemic data, outperforming pairwise methods in detecting complex group interactions.

ABSTRACT

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.

Motivation & Objective

• To address the limitation of pairwise statistical methods in capturing group interactions in multivariate time series.
• To develop a framework that identifies and quantifies higher-order dependencies beyond pairwise correlations.
• To apply topological data analysis and network theory to detect persistent patterns of co-fluctuation in real-world complex systems.
• To validate the method on simulated chaotic systems and real data from neuroscience, finance, and epidemiology.
• To demonstrate that higher-order structures reveal dynamical regimes and functional organization not detectable via standard functional connectivity.

Proposed method

• The method constructs a hypergraph from time series data by identifying co-fluctuating groups of variables using a sliding window approach.
• It computes a hypergraph's hyper-coherence and hyper-complexity to quantify the strength and persistence of higher-order interactions.
• The framework uses persistent homology to detect topological features such as 1D loops (holes) in the space of co-fluctuations, representing persistent group dynamics.
• Violating triangles (inconsistent triplets) are identified as indicators of non-trivial higher-order structure in the data.
• The method applies a thresholding strategy to isolate extreme frames—either the most hyper-coherent or least hyper-complex—for detailed analysis.
• It integrates local higher-order indicators (e.g., nodal and edge-level hyper-coherence and homological persistence) to map functional networks at the level of brain regions.

Experimental results

Research questions

• RQ1 Can higher-order topological structures in multivariate time series reveal dynamical regimes not detectable by pairwise statistics?
• RQ2 How do persistent topological features such as 1D holes correlate with known functional brain networks like the Default Mode Network?
• RQ3 To what extent can higher-order indicators improve classification of complex system states, such as disease outbreaks or financial stress?
• RQ4 Do brain functional networks exhibit consistent higher-order co-fluctuation patterns across different thresholds of hyper-coherence and hyper-complexity?
• RQ5 Can the proposed method distinguish between individual and collective dynamics in real-world time series from neuroscience, finance, and epidemiology?

Key findings

• The method successfully differentiates chaotic dynamical phases and synchronization types in coupled chaotic maps, outperforming pairwise methods.
• In brain data, the most hyper-coherent frames consistently activate the Default Mode Network and sensorimotor regions, with topological features persisting across threshold levels.
• The homological scaffold reveals persistent 1D holes concentrated in the Default Mode Network and FrontoParietal network, indicating their role in integrating higher-order dynamics.
• For US historical disease data, the RBF SVM classifier achieved an average accuracy of 0.85 using hyper-coherence, edge violations, and hyper-complexity as features.
• Random Forest and RBF SVM achieved the highest F1 weighted scores of 0.85, demonstrating the predictive power of higher-order indicators.
• The framework identifies robust, reproducible signatures of higher-order organization across diverse systems, validating its utility in complex systems analysis.