Beyond Spatio-Temporal Representations: Evolving Fourier Transform for Temporal Graphs

TL;DR

This work introduces the Evolving Graph Fourier Transform (EFT), an invertible spectral transform that jointly captures evolving spectra on temporal graphs by optimizing over the joint Laplacian $L_{C\mathcal{J_D}}$ with pseudospectrum relaxations. EFT is realized as a Kronecker-structured transform combining a time Fourier basis $\boldsymbol{\Psi_T}$ and per-time graph bases $\boldsymbol{\Psi_{G_t}}$, enabling efficient, interpretable frequency analysis across time and graph domains while reducing computational complexity relative to joint eigendecomposition. An EFT-based transformer variant (EFT-T) demonstrates state-of-the-art performance on large-scale continuous-time and standard discrete temporal graphs, with notable denoising capabilities and ablation-backed importance of both graph- and time-domain filtering. Theoretical bounds relate EFT to the absolute decomposition (AD), showing the transform remains close to the exact joint eigensolution under bounded graph evolution, and practical results confirm EFT's robustness and scalability for dynamic graph representation learning.

Abstract

We present the Evolving Graph Fourier Transform (EFT), the first invertible spectral transform that captures evolving representations on temporal graphs. We motivate our work by the inadequacy of existing methods for capturing the evolving graph spectra, which are also computationally expensive due to the temporal aspect along with the graph vertex domain. We view the problem as an optimization over the Laplacian of the continuous time dynamic graph. Additionally, we propose pseudo-spectrum relaxations that decompose the transformation process, making it highly computationally efficient. The EFT method adeptly captures the evolving graph's structural and positional properties, making it effective for downstream tasks on evolving graphs. Hence, as a reference implementation, we develop a simple neural model induced with EFT for capturing evolving graph spectra. We empirically validate our theoretical findings on a number of large-scale and standard temporal graph benchmarks and demonstrate that our model achieves state-of-the-art performance.

Figures (6)

• Figure 1: Left circular figure shows equivalence between EFT and existing transformations (DFT sundararajan2023discrete, JFT loukas2016frequency, GFT ortega2018graph). Each directed arrow (e.g, A to B), interprets as a transform simulation (transform A can be simulated by B using edge annotations). Right part shows timestamp-wise product between signals and graph structure. Here, nodes of next timestep are connected by dotted arrows to obtain the graph $\mathcal{J_D}$ which can be used by GFT to simulate EFT (if graph is static).
• Figure 2: Reconstruction error on noisy synthetic data.
• Figure 3: Representations on dynamic mesh datasets. Left (a,b): Reconstruction error on the datasets illustrating the compactness of EFT . Right (c): Illustration of filtering using EFT on the dynamic mesh of a Dancer.
• Figure 4: Effect of inducing 1) semantic noise in embeddings with and without filters (a-b) 2) structural noise in the form of graph perturbations with and without graph filters (c-d), on the performance of EFT . We consider large-scale SR setting.
• Figure 5: Filter frequency responses learnt by EFT on dynamic graph datasets. The x-axis shows the vertex frequency (0-2), y axis shows the normalized temporal frequency and z axis shows the magnitudes of the normalized frequency response.

Theorems & Definitions (14)

• Lemma 4.1
• Lemma 4.2
• Theorem 5.1
• Lemma B.1
• proof
• Lemma B.2
• proof
• Remark B.1
• proof
• Theorem B.1