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Continuous Product Graph Neural Networks

Aref Einizade, Fragkiskos D. Malliaros, Jhony H. Giraldo

TL;DR

CITRUS leverages the separability of continuous heat kernels from Cartesian graph products to efficiently implement graph spectral decomposition and is evaluated on well-known traffic and weather spatiotemporal forecasting datasets, demonstrating superior performance over existing approaches.

Abstract

Processing multidomain data defined on multiple graphs holds significant potential in various practical applications in computer science. However, current methods are mostly limited to discrete graph filtering operations. Tensorial partial differential equations on graphs (TPDEGs) provide a principled framework for modeling structured data across multiple interacting graphs, addressing the limitations of the existing discrete methodologies. In this paper, we introduce Continuous Product Graph Neural Networks (CITRUS) that emerge as a natural solution to the TPDEG. CITRUS leverages the separability of continuous heat kernels from Cartesian graph products to efficiently implement graph spectral decomposition. We conduct thorough theoretical analyses of the stability and over-smoothing properties of CITRUS in response to domain-specific graph perturbations and graph spectra effects on the performance. We evaluate CITRUS on well-known traffic and weather spatiotemporal forecasting datasets, demonstrating superior performance over existing approaches. The implementation codes are available at https://github.com/ArefEinizade2/CITRUS.

Continuous Product Graph Neural Networks

TL;DR

CITRUS leverages the separability of continuous heat kernels from Cartesian graph products to efficiently implement graph spectral decomposition and is evaluated on well-known traffic and weather spatiotemporal forecasting datasets, demonstrating superior performance over existing approaches.

Abstract

Processing multidomain data defined on multiple graphs holds significant potential in various practical applications in computer science. However, current methods are mostly limited to discrete graph filtering operations. Tensorial partial differential equations on graphs (TPDEGs) provide a principled framework for modeling structured data across multiple interacting graphs, addressing the limitations of the existing discrete methodologies. In this paper, we introduce Continuous Product Graph Neural Networks (CITRUS) that emerge as a natural solution to the TPDEG. CITRUS leverages the separability of continuous heat kernels from Cartesian graph products to efficiently implement graph spectral decomposition. We conduct thorough theoretical analyses of the stability and over-smoothing properties of CITRUS in response to domain-specific graph perturbations and graph spectra effects on the performance. We evaluate CITRUS on well-known traffic and weather spatiotemporal forecasting datasets, demonstrating superior performance over existing approaches. The implementation codes are available at https://github.com/ArefEinizade2/CITRUS.
Paper Structure (29 sections, 13 theorems, 36 equations, 5 figures, 9 tables)

This paper contains 29 sections, 13 theorems, 36 equations, 5 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Methodology and Theoretical Analysis
  4. Continuous Product Graph Neural Networks (CITRUS)
  5. Stability Analysis
  6. Over-smoothing Analysis
  7. Experimental Results
  8. Experimental Stability Analysis
  9. Experimental Over-smoothing Analysis
  10. Experiments on Real-world Data
  11. Ablation Study and Hyperparameter Sensitivity Analysis
  12. Conclusion and Limitations
  13. Proofs
  14. Proof of Theorem \ref{['MDPDE_thm']}
  15. Proof of Theorem \ref{['MDPDE_theorem']}
  16. ...and 14 more sections

Key Result

Theorem 3.2

Let $\tilde{\underline{\mathbf{U}}}_0$ be the the initial value of $\tilde{\underline{\mathbf{U}}}_t$. The solution to the TPDEG in MDPDE is given by:

Figures (5)

  • Figure 1: Illustration of key concepts of CITRUS. a) Cartesian product between three-factor graphs. b) Continous product graph function (CITRUS) operating on the multidomain graph data $\underline{\mathbf{U}}$.
  • Figure 2: Stability analysis vs. different SNR scenarios, related the results in Theorem \ref{['thm_e1e2']}.
  • Figure 3: Over-smoothing analysis using Theorem \ref{['exp_ener_prod']}. left:$\ln{s}-\frac{2}{P}\tilde{t}\tilde{\lambda}<0$, right:$\ln{s}-\frac{2}{P}\tilde{t}\tilde{\lambda}>0$.
  • Figure 4: Explained variance ratio vs. selected principal components.
  • Figure 5: Log relative distance vs. increasing the number of layers for different values of $t$ in actual bounds.

Theorems & Definitions (25)

  • Definition 3.1: Tensorial PDE on graphs (TPDEG)
  • Theorem 3.2
  • Proposition 3.3
  • Remark 3.4
  • Remark 3.5
  • Proposition 3.6
  • Theorem 3.7
  • Definition 3.8
  • Lemma 3.9
  • Theorem 3.10
  • ...and 15 more