[Paper Review] Z-GCNETs: Time Zigzags at Graph Convolutional Networks for Time Series Forecasting

Yuzhou Chen, Ignacio Segovia-Domínguez+1 more|arXiv (Cornell University)|May 10, 2021

Topological and Geometric Data AnalysisComputer Science39 references44 citations

TL;DR

Z-GCNETs introduces a time-aware zigzag persistence layer inside graph convolutional networks to leverage time-conditioned topological features for improved time series forecasting on traffic and Ethereum token networks. It outperforms 13 state-of-the-art methods on 4 datasets."

ABSTRACT

There recently has been a surge of interest in developing a new class of deep learning (DL) architectures that integrate an explicit time dimension as a fundamental building block of learning and representation mechanisms. In turn, many recent results show that topological descriptors of the observed data, encoding information on the shape of the dataset in a topological space at different scales, that is, persistent homology of the data, may contain important complementary information, improving both performance and robustness of DL. As convergence of these two emerging ideas, we propose to enhance DL architectures with the most salient time-conditioned topological information of the data and introduce the concept of zigzag persistence into time-aware graph convolutional networks (GCNs). Zigzag persistence provides a systematic and mathematically rigorous framework to track the most important topological features of the observed data that tend to manifest themselves over time. To integrate the extracted time-conditioned topological descriptors into DL, we develop a new topological summary, zigzag persistence image, and derive its theoretical stability guarantees. We validate the new GCNs with a time-aware zigzag topological layer (Z-GCNETs), in application to traffic forecasting and Ethereum blockchain price prediction. Our results indicate that Z-GCNET outperforms 13 state-of-the-art methods on 4 time series datasets.

Motivation & Objective

• Motivate integrating explicit time-conditioned representations with topological data analysis in deep learning.
• Develop zigzag persistence-based topological summaries (zigzag persistence images) with stability guarantees.
• Incorporate time-aware topological features into GCNs via a dedicated zigzag layer (Z-GCNETs).
• Fuse spatial, temporal, and topological information within a unified forecasting framework.

Proposed method

• Construct time-evolving graphs from multivariate time series data with sliding windows.
• Extract zigzag persistence diagrams across time and convert them into zigzag persistence images (ZPI) with a stability guarantee.
• Process ZPI through CNNs to obtain a learned topological representation (Z^{( abla)}).
• Integrate Z^{( abla)} with spatial graph convolution using a diffusion-like Laplacianlink (L̃) and a recurrent module (GRU) for multi-step forecasting.
• Employ a time-aware zigzag topological layer to fuse spatial, temporal, and topological features before GRU-based prediction.

Experimental results

Research questions

• RQ1 Can time-conditioned zigzag persistence improve predictive accuracy for time series on graphs beyond standard spatial-temporal GCNs?
• RQ2 What is the stability and practical benefit of representing zigzag persistence as images within a neural network?
• RQ3 How does incorporating topological information affect robustness and performance across traffic and blockchain datasets?

Key findings

• Z-GCNETs outperform 13 state-of-the-art baselines on four time series datasets.
• On PeMSD4/PeMSD8, Z-GCNETs achieve lower RMSE/MAE/MAPE than competitors, e.g., PeMSD4 RMSE 31.61 vs. 32.30–39.88 for others.
• On Ethereum token networks (Bytom, Decentraland), Z-GCNETs achieve better MAPE (31.04% and 23.81%) than baselines.
• Ablation shows removing zigzag learning, spatial GCN, or temporal GCN degrades performance, with temporal GCN being particularly impactful.
• ZPI representations are stable with respect to the Wasserstein-1 distance, ensuring reliable topological summaries.