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Enhancing Topological Dependencies in Spatio-Temporal Graphs with Cycle Message Passing Blocks

Minho Lee, Yun Young Choi, Sun Woo Park, Seunghwan Lee, Joohwan Ko, Jaeyoung Hong

TL;DR

The paper tackles spatio-temporal forecasting on traffic networks, addressing limitations of traditional MPNN/Transformer approaches that encode temporal and spatial relations separately. It introduces Cy2Mixer, a three-block spatio-temporal GNN leveraging a cycle message-passing block and clique adjacency (A_C) to encode topological invariants, underpinned by a mathematical basis linking temporal cycles to cycle bases. The authors provide theoretical grounding via topological invariants and demonstrate state-of-the-art or competitive results across six traffic datasets, plus an extension to air-pollution prediction, while highlighting computational efficiency over DTW-based methods. The work suggests that incorporating topological structure—particularly cyclic subgraphs—can yield meaningful gains in forecasting accuracy for networks with rich topology, with practical implications for transportation and beyond.

Abstract

Graph Neural Networks (GNNs) and Transformer-based models have been increasingly adopted to learn the complex vector representations of spatio-temporal graphs, capturing intricate spatio-temporal dependencies crucial for applications such as traffic datasets. Although many existing methods utilize multi-head attention mechanisms and message-passing neural networks (MPNNs) to capture both spatial and temporal relations, these approaches encode temporal and spatial relations independently, and reflect the graph's topological characteristics in a limited manner. In this work, we introduce the Cycle to Mixer (Cy2Mixer), a novel spatio-temporal GNN based on topological non-trivial invariants of spatio-temporal graphs with gated multi-layer perceptrons (gMLP). The Cy2Mixer is composed of three blocks based on MLPs: A temporal block for capturing temporal properties, a message-passing block for encapsulating spatial information, and a cycle message-passing block for enriching topological information through cyclic subgraphs. We bolster the effectiveness of Cy2Mixer with mathematical evidence emphasizing that our cycle message-passing block is capable of offering differentiated information to the deep learning model compared to the message-passing block. Furthermore, empirical evaluations substantiate the efficacy of the Cy2Mixer, demonstrating state-of-the-art performances across various spatio-temporal benchmark datasets. The source code is available at https://github.com/leemingo/cy2mixer.

Enhancing Topological Dependencies in Spatio-Temporal Graphs with Cycle Message Passing Blocks

TL;DR

The paper tackles spatio-temporal forecasting on traffic networks, addressing limitations of traditional MPNN/Transformer approaches that encode temporal and spatial relations separately. It introduces Cy2Mixer, a three-block spatio-temporal GNN leveraging a cycle message-passing block and clique adjacency (A_C) to encode topological invariants, underpinned by a mathematical basis linking temporal cycles to cycle bases. The authors provide theoretical grounding via topological invariants and demonstrate state-of-the-art or competitive results across six traffic datasets, plus an extension to air-pollution prediction, while highlighting computational efficiency over DTW-based methods. The work suggests that incorporating topological structure—particularly cyclic subgraphs—can yield meaningful gains in forecasting accuracy for networks with rich topology, with practical implications for transportation and beyond.

Abstract

Graph Neural Networks (GNNs) and Transformer-based models have been increasingly adopted to learn the complex vector representations of spatio-temporal graphs, capturing intricate spatio-temporal dependencies crucial for applications such as traffic datasets. Although many existing methods utilize multi-head attention mechanisms and message-passing neural networks (MPNNs) to capture both spatial and temporal relations, these approaches encode temporal and spatial relations independently, and reflect the graph's topological characteristics in a limited manner. In this work, we introduce the Cycle to Mixer (Cy2Mixer), a novel spatio-temporal GNN based on topological non-trivial invariants of spatio-temporal graphs with gated multi-layer perceptrons (gMLP). The Cy2Mixer is composed of three blocks based on MLPs: A temporal block for capturing temporal properties, a message-passing block for encapsulating spatial information, and a cycle message-passing block for enriching topological information through cyclic subgraphs. We bolster the effectiveness of Cy2Mixer with mathematical evidence emphasizing that our cycle message-passing block is capable of offering differentiated information to the deep learning model compared to the message-passing block. Furthermore, empirical evaluations substantiate the efficacy of the Cy2Mixer, demonstrating state-of-the-art performances across various spatio-temporal benchmark datasets. The source code is available at https://github.com/leemingo/cy2mixer.
Paper Structure (41 sections, 1 theorem, 12 equations, 6 figures, 8 tables)

This paper contains 41 sections, 1 theorem, 12 equations, 6 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. Preliminaries
  4. Message passing neural networks (MPNNs)
  5. gated Multi-Layer Perceptron (gMLP)
  6. Traffic Prediction Problem
  7. Traffic sensor
  8. Traffic network
  9. Traffic signal
  10. Problem formalization
  11. Mathematical Backgrounds
  12. Topological Space
  13. Temporal cyclic structures
  14. Temporal cliques
  15. Methodology
  16. ...and 26 more sections

Key Result

Theorem 4.1

Given any choice of $t_0 \in I$, let $\pi_{t_0}: \mathcal{G} \times I \to \mathcal{G} \times \{t_0\} \cong \mathcal{G}$ be the projection map which sends the interval $I$ to a singleton set $\{t_0\}$. Let $\mathcal{C}_{\mathcal{G} \times I} := \{C_1, C_2, \cdots, C_n\}$ be a cycle basis of the topol is a cycle basis of the traffic network $\mathcal{G}$.

Figures (6)

  • Figure 1: Prediction example of Cy2Mixer in the PEMS04 dataset. While Cy2Mixer shows similar performance between Node 170 and Node 173, which are connected via the adjacency matrix $A$, Cy2Mixer exhibits superior performance at Node 0 when utilizing the clique adjacency matrix $A_C$, indicating the effectiveness of the cycle block in capturing cyclic subgraph relationships.
  • Figure 2: An illustration of lifting a cyclic subgraph of the traffic network $\mathcal{G}$ to a temporal cyclic subgraph of the topological space $\mathcal{G} \times I$ representing the traffic dataset. Every temporal cyclic subgraph of the traffic dataset $\mathcal{G} \times I$ can be obtained from cyclic subgraph of the underlying traffic network $\mathcal{G}$ by using Theorem \ref{['theorem:cycle_temporal']}. By transforming cyclic subgraphs into cliques by adding and deleting edges suitably, Cy2Mixer effectively models how traffic signals measured in distinct nodes and time can affect each other.
  • Figure 3: The overall framework of Cy2Mixer. Layers within the green box indicate the Cy2Mixer encoder layer, which comprises a temporal block, spatial message-passing block, and cycle message-passing block to ensure a comprehensive understanding of both temporal and spatial aspects.
  • Figure 4: Three matrices of PEMS04: (a) Adjacency matrix $A$, (b) Clique adjacency matrix $A_{C}$, and (c) Matrix constructed by DTW algorithm, respectively. The figure demonstrates that the clique adjacency matrix we use can capture topological information different from the standard adjacency matrix and DTW matrix.
  • Figure 5: Visualization of prediction results for Node 170, Node 173 (connected to Node 170 in the adjacency matrix), and Nodes 0 and 169 (connected to Node 170 in the clique adjacency matrix) in the PEMS04 dataset.
  • ...and 1 more figures

Theorems & Definitions (1)

  • Theorem 4.1