# Z-GCNETs: Time Zigzags at Graph Convolutional Networks for Time Series Forecasting

arXiv: https://arxiv.org/abs/2105.04100 · HTML: https://mlsanity.com/paper/2105.04100/

## Key facts

- Cited by 67 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 51 references that are in the map.
- First posted on arXiv 10 May 2021.

## Builds on (most cited first, 23 of 23)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Sequence to Sequence Learning with Neural Networks](https://mlsanity.com/paper/1409.3215/) (2014)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Spatial Temporal Graph Convolutional Networks for Skeleton-Based Action Recognition](https://mlsanity.com/paper/1801.07455/) (2018)
- [Spatio-Temporal Graph Convolutional Networks: A Deep Learning Framework for Traffic Forecasting](https://mlsanity.com/paper/1709.04875/) (2017)
- [Graph WaveNet for Deep Spatial-Temporal Graph Modeling](https://mlsanity.com/paper/1906.00121/) (2019)
- [Adaptive Graph Convolutional Recurrent Network for Traffic Forecasting](https://mlsanity.com/paper/2007.02842/) (2020)
- [EvolveGCN: Evolving Graph Convolutional Networks for Dynamic Graphs](https://mlsanity.com/paper/1902.10191/) (2019)
- [LSTM: A Search Space Odyssey](https://mlsanity.com/paper/1503.04069/) (2015)
- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [Anti-Money Laundering in Bitcoin: Experimenting with Graph Convolutional Networks for Financial Forensics](https://mlsanity.com/paper/1908.02591/) (2019)
- [Deep Multi-View Spatial-Temporal Network for Taxi Demand Prediction](https://mlsanity.com/paper/1802.08714/) (2018)
- [PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures](https://mlsanity.com/paper/1904.09378/) (2019)
- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [Topological Approaches to Deep Learning](https://mlsanity.com/paper/1811.01122/) (2018)
- [Dissecting Ethereum Blockchain Analytics: What We Learn from Topology and Geometry of Ethereum Graph](https://mlsanity.com/paper/1912.10105/) (2019)
- [Capturing Dynamics of Time-Varying Data via Topology](https://mlsanity.com/paper/2010.05780/) (2020)
- [Blockchain Networks: Data Structures of Bitcoin, Monero, Zcash, Ethereum, Ripple and Iota](https://mlsanity.com/paper/2103.08712/) (2021)
- [Evasion Paths in Mobile Sensor Networks](https://mlsanity.com/paper/1308.3536/) (2013)
- [Coordinate-Free Quantification of Coverage in Dynamic Sensor Networks](https://mlsanity.com/paper/1411.7337/) (2014)
- [The Importance of Forgetting: Limiting Memory Improves Recovery of Topological Characteristics from Neural Data](https://mlsanity.com/paper/1710.11279/) (2017)

## Built on by (most cited first, 30 of 67)

- [A Time Series is Worth 64 Words: Long-term Forecasting with Transformers](https://mlsanity.com/paper/2211.14730/) (2022)
- [A Survey on Graph Neural Networks for Time Series: Forecasting, Classification, Imputation, and Anomaly Detection](https://mlsanity.com/paper/2307.03759/) (2023)
- [Graph Neural Controlled Differential Equations for Traffic Forecasting](https://mlsanity.com/paper/2112.03558/) (2021)
- [Exploring Progress in Multivariate Time Series Forecasting: Comprehensive Benchmarking and Heterogeneity Analysis](https://mlsanity.com/paper/2310.06119/) (2023)
- [Networked Time Series Imputation via Position-aware Graph Enhanced Variational Autoencoders](https://mlsanity.com/paper/2305.18612/) (2023)
- [Spatial-Temporal-Decoupled Masked Pre-training for Spatiotemporal Forecasting](https://mlsanity.com/paper/2312.00516/) (2023)
- [Spatio-Temporal Joint Graph Convolutional Networks for Traffic Forecasting](https://mlsanity.com/paper/2111.13684/) (2021)
- [Regularized Graph Structure Learning with Semantic Knowledge for Multi-variates Time-Series Forecasting](https://mlsanity.com/paper/2210.06126/) (2022)
- [ST-MLP: A Cascaded Spatio-Temporal Linear Framework with Channel-Independence Strategy for Traffic Forecasting](https://mlsanity.com/paper/2308.07496/) (2023)
- [Do We Really Need Graph Neural Networks for Traffic Forecasting?](https://mlsanity.com/paper/2301.12603/) (2023)
- [Building Transportation Foundation Model via Generative Graph Transformer](https://mlsanity.com/paper/2305.14826/) (2023)
- [Graph Neural Rough Differential Equations for Traffic Forecasting](https://mlsanity.com/paper/2303.10909/) (2023)
- [Localised Adaptive Spatial-Temporal Graph Neural Network](https://mlsanity.com/paper/2306.06930/) (2023)
- [FRIGATE: Frugal Spatio-temporal Forecasting on Road Networks](https://mlsanity.com/paper/2306.08277/) (2023)
- [Dynamic Hypergraph Structure Learning for Traffic Flow Forecasting](https://mlsanity.com/paper/2309.12028/) (2023)
- [A Survey of Generative Techniques for Spatial-Temporal Data Mining](https://mlsanity.com/paper/2405.09592/) (2024)
- [PGCN: Progressive Graph Convolutional Networks for Spatial-Temporal Traffic Forecasting](https://mlsanity.com/paper/2202.08982/) (2022)
- [ToDD: Topological Compound Fingerprinting in Computer-Aided Drug Discovery](https://mlsanity.com/paper/2211.03808/) (2022)
- [Explainable Spatio-Temporal Graph Neural Networks](https://mlsanity.com/paper/2310.17149/) (2023)
- [STAEformer: Spatio-Temporal Adaptive Embedding Makes Vanilla Transformer SOTA for Traffic Forecasting](https://mlsanity.com/paper/2308.10425/) (2023)
- [Efficient Traffic Prediction Through Spatio-Temporal Distillation](https://mlsanity.com/paper/2501.10459/) (2025)
- [Enhancing the Robustness via Adversarial Learning and Joint Spatial-Temporal Embeddings in Traffic Forecasting](https://mlsanity.com/paper/2208.03063/) (2022)
- [Spatio-Temporal Meta Contrastive Learning](https://mlsanity.com/paper/2310.17678/) (2023)
- [A Systematic Literature Review of Spatio-Temporal Graph Neural Network Models for Time Series Forecasting and Classification](https://mlsanity.com/paper/2410.22377/) (2024)
- [Learnable Path in Neural Controlled Differential Equations](https://mlsanity.com/paper/2301.04333/) (2023)
- [Hybrid Variational Autoencoder for Time Series Forecasting](https://mlsanity.com/paper/2303.07048/) (2023)
- [Easy Begun is Half Done: Spatial-Temporal Graph Modeling with ST-Curriculum Dropout](https://mlsanity.com/paper/2211.15182/) (2022)
- [SpoT-Mamba: Learning Long-Range Dependency on Spatio-Temporal Graphs with Selective State Spaces](https://mlsanity.com/paper/2406.11244/) (2024)
- [Continual Traffic Forecasting via Mixture of Experts](https://mlsanity.com/paper/2406.03140/) (2024)
- [Attention-based Spatial-Temporal Graph Convolutional Recurrent Networks for Traffic Forecasting](https://mlsanity.com/paper/2302.12973/) (2023)

## Abstract (arXiv)

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.