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Expressive Power of Temporal Message Passing

Przemysław Andrzej Wałęga, Michael Rawson

TL;DR

This work rigorously analyzes the expressive power of temporal message-passing graph neural networks by partitioning them into global and local classes. It introduces a Weisfeiler-Leman–based framework, transforming temporal graphs into two knowledge graphs, $\mathcal{K}_{\mathsf{glob}}(TG)$ and $\mathcal{K}_{\mathsf{loc}}(TG)$, to exactly characterise node distinguishability via $1$-WL. The key results show that global and local MP-TGNNs have incomparable expressive power in general, but on colour-persistent graphs, local models are strictly more expressive; these insights are corroborated by experiments on the Temporal Graph Benchmark 2.0, where local models outperform global ones under matched conditions. The findings provide principled guidance for selecting temporal message-passing schemes and for designing new models with desired discriminative capabilities. Overall, the paper bridges theory and practice, offering a robust toolkit for understanding and leveraging temporal expressive power in TGNNs.

Abstract

Graph neural networks (GNNs) have recently been adapted to temporal settings, often employing temporal versions of the message-passing mechanism known from GNNs. We divide temporal message passing mechanisms from literature into two main types: global and local, and establish Weisfeiler-Leman characterisations for both. This allows us to formally analyse expressive power of temporal message-passing models. We show that global and local temporal message-passing mechanisms have incomparable expressive power when applied to arbitrary temporal graphs. However, the local mechanism is strictly more expressive than the global mechanism when applied to colour-persistent temporal graphs, whose node colours are initially the same in all time points. Our theoretical findings are supported by experimental evidence, underlining practical implications of our analysis.

Expressive Power of Temporal Message Passing

TL;DR

This work rigorously analyzes the expressive power of temporal message-passing graph neural networks by partitioning them into global and local classes. It introduces a Weisfeiler-Leman–based framework, transforming temporal graphs into two knowledge graphs, and , to exactly characterise node distinguishability via -WL. The key results show that global and local MP-TGNNs have incomparable expressive power in general, but on colour-persistent graphs, local models are strictly more expressive; these insights are corroborated by experiments on the Temporal Graph Benchmark 2.0, where local models outperform global ones under matched conditions. The findings provide principled guidance for selecting temporal message-passing schemes and for designing new models with desired discriminative capabilities. Overall, the paper bridges theory and practice, offering a robust toolkit for understanding and leveraging temporal expressive power in TGNNs.

Abstract

Graph neural networks (GNNs) have recently been adapted to temporal settings, often employing temporal versions of the message-passing mechanism known from GNNs. We divide temporal message passing mechanisms from literature into two main types: global and local, and establish Weisfeiler-Leman characterisations for both. This allows us to formally analyse expressive power of temporal message-passing models. We show that global and local temporal message-passing mechanisms have incomparable expressive power when applied to arbitrary temporal graphs. However, the local mechanism is strictly more expressive than the global mechanism when applied to colour-persistent temporal graphs, whose node colours are initially the same in all time points. Our theoretical findings are supported by experimental evidence, underlining practical implications of our analysis.
Paper Structure (33 sections, 19 theorems, 29 equations, 14 figures, 3 tables)

This paper contains 33 sections, 19 theorems, 29 equations, 14 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Background
  4. Temporal graphs.
  5. Temporal graph neural networks with message-passing.
  6. Weisfeiler-Leman algorithm.
  7. Related Work
  8. Temporal Weisfeiler-Leman Characterisation
  9. Timewise Isomorphism
  10. Relative Expressiveness of Temporal Message Passing Mechanisms
  11. Experiments
  12. Benchmarks.
  13. Models.
  14. Implementation.
  15. Training
  16. ...and 18 more sections

Key Result

Theorem 2

For any temporal graph $TG$, any timestamped nodes $(v,t)$ and $(u,t')$ in $TG$, and any $\ell \in \mathbb{N}$:

Figures (14)

  • Figure 1: Our approach to determine which nodes in a temporal graph $TG$ are distinguishable by $\mathsf{MP}\text{-}\mathsf{TGNN}$: we construct of knowledge graphs $\mathcal{K}_{\mathsf{glob}}(TG)$ and $\mathcal{K}_{\mathsf{loc}}(TG)$, and then apply 1-WL
  • Figure 3: A temporal graph in the snapshot representation
  • Figure 4: A colour-persistent temporal graph (\ref{['fig:snap']}) and its aggregated representation (\ref{['fig:aggr']})
  • Figure 5: $\mathcal{K}_{\mathsf{glob}}(TG)$ constructed for $TG$ from \ref{['fig:snapshot']}
  • Figure 6: Knowledge graph $\mathcal{K}_{\mathsf{loc}}(TG)$ for $TG$ from \ref{['fig:snapshot']}
  • ...and 9 more figures

Theorems & Definitions (40)

  • Definition 1
  • Theorem 2
  • proof : Proof sketch
  • Theorem 3
  • proof : Proof sketch
  • Definition 4
  • Theorem 5
  • Theorem 6
  • Definition 7
  • Theorem 8
  • ...and 30 more