Persistent Homology-induced Graph Ensembles for Time Series Regressions

TL;DR

The paper tackles the dependency on fixed input graphs in spatio-temporal graph neural networks for time-series tasks by introducing persistent-homology–based graphs that capture multiscale topology. It builds multiple graphs from Vietoris–Rips filtrations at death times $\tau_i^d$ and processes each with its own simple encoder, fusing them through an attention-based ensemble of GNNs. Theoretical analysis argues that these PH-induced graphs preserve information at topology-changing thresholds and that instance-dependent fusion further enhances predictive power, while experiments on seismic TSER and traffic forecasting show consistent improvements over baselines. This approach provides a principled, interpretable way to leverage multiscale geometric structure in sensor networks and suggests future work on single-graph representations via optimal transport barycenters.

Abstract

The effectiveness of Spatio-temporal Graph Neural Networks (STGNNs) in time-series applications is often limited by their dependence on fixed, hand-crafted input graph structures. Motivated by insights from the Topological Data Analysis (TDA) paradigm, of which real-world data exhibits multi-scale patterns, we construct several graphs using Persistent Homology Filtration -- a mathematical framework describing the multiscale structural properties of data points. Then, we use the constructed graphs as an input to create an ensemble of Graph Neural Networks. The ensemble aggregates the signals from the individual learners via an attention-based routing mechanism, thus systematically encoding the inherent multiscale structures of data. Four different real-world experiments on seismic activity prediction and traffic forecasting (PEMS-BAY, METR-LA) demonstrate that our approach consistently outperforms single-graph baselines while providing interpretable insights.

Figures (4)

• Figure 1: Computation step of persistent homology given data points. (a) A filtration that tracks the topological changes of the dataset with two homology classes $H_0$ (connected components) and $H_1$ (loops or tunnels). (b) The persistent barcode of the corresponding filtration. Epsilon ($\epsilon$) is the radius parameter that determines when points are connected - as it increases, points within $\epsilon$ distance of each other become connected, revealing the data's topological structure (c) The persistence diagram shows the lifespan of topological features---an equivalent representation of the barcode.
• Figure 2: Graph generation using persistent homology. Each graph is generated by the filtration at the threshold $\epsilon_t$. The graphs are fed separately into arbitrary functions $f_{\theta_k}$ parametrized by learnable weights $\theta_k$ (e.g., neural networks), which are then aggregated ($AGG$) to obtain the representations $\mathbf{\overline{Z}}$ for the downstream tasks.
• Figure 3: Results of varying window times as input.
• Figure 4: Graph structures on CW dataset. Three highest-weighted networks with their corresponding death times.