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LEAP: Local ECT-Based Learnable Positional Encodings for Graphs

Juan Amboage, Ernst Röell, Patrick Schnider, Bastian Rieck

TL;DR

LEAP introduces a local, learnable positional encoding for graphs based on the differentiable local Euler Characteristic Transform ($\ell$-ECT). By normalizing $m$-hop neighborhoods and computing the ECT over a grid of directions and thresholds, LEAP produces a flexible, end-to-end trainable graph embedding, with multiple projection strategies to obtain a compact $k$-D vector. Experiments on synthetic and real-world benchmarks show LEAP, especially with learnable directions, consistently improves performance over standard baselines and static PEs, and demonstrate LEAP’s ability to capture topological structure even when node features are uninformative. The approach highlights the potential of topological encodings as powerful components in graph representation learning and lays groundwork for further integration with global PEs and higher-order topology.

Abstract

Graph neural networks (GNNs) largely rely on the message-passing paradigm, where nodes iteratively aggregate information from their neighbors. Yet, standard message passing neural networks (MPNNs) face well-documented theoretical and practical limitations. Graph positional encoding (PE) has emerged as a promising direction to address these limitations. The Euler Characteristic Transform (ECT) is an efficiently computable geometric-topological invariant that characterizes shapes and graphs. In this work, we combine the differentiable approximation of the ECT (DECT) and its local variant ($\ell$-ECT) to propose LEAP, a new end-to-end trainable local structural PE for graphs. We evaluate our approach on multiple real-world datasets as well as on a synthetic task designed to test its ability to extract topological features. Our results underline the potential of LEAP-based encodings as a powerful component for graph representation learning pipelines.

LEAP: Local ECT-Based Learnable Positional Encodings for Graphs

TL;DR

LEAP introduces a local, learnable positional encoding for graphs based on the differentiable local Euler Characteristic Transform (-ECT). By normalizing -hop neighborhoods and computing the ECT over a grid of directions and thresholds, LEAP produces a flexible, end-to-end trainable graph embedding, with multiple projection strategies to obtain a compact -D vector. Experiments on synthetic and real-world benchmarks show LEAP, especially with learnable directions, consistently improves performance over standard baselines and static PEs, and demonstrate LEAP’s ability to capture topological structure even when node features are uninformative. The approach highlights the potential of topological encodings as powerful components in graph representation learning and lays groundwork for further integration with global PEs and higher-order topology.

Abstract

Graph neural networks (GNNs) largely rely on the message-passing paradigm, where nodes iteratively aggregate information from their neighbors. Yet, standard message passing neural networks (MPNNs) face well-documented theoretical and practical limitations. Graph positional encoding (PE) has emerged as a promising direction to address these limitations. The Euler Characteristic Transform (ECT) is an efficiently computable geometric-topological invariant that characterizes shapes and graphs. In this work, we combine the differentiable approximation of the ECT (DECT) and its local variant (-ECT) to propose LEAP, a new end-to-end trainable local structural PE for graphs. We evaluate our approach on multiple real-world datasets as well as on a synthetic task designed to test its ability to extract topological features. Our results underline the potential of LEAP-based encodings as a powerful component for graph representation learning pipelines.

Paper Structure

This paper contains 25 sections, 1 theorem, 5 equations, 6 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Message Passing
  4. Random Walk Positional Encoding (RWPE).
  5. Laplacian Positional Encoding (LaPE).
  6. The Euler Characteristic Transform (ECT)
  7. Methods
  8. l-ECT based Positional Encoding
  9. ECT projection strategies
  10. Experiments
  11. Synthetic Dataset
  12. Classifying real-world datasets
  13. Architectures.
  14. Experimental setup.
  15. Ablations
  16. ...and 10 more sections

Key Result

Theorem 1

Let $({\mathcal{G}}, x)$ be a featured graph, and let $\{\ell\textnormal{-ECT}_1[{\mathcal{G}},x; v]\}_v$ be the set of the 1-hop $\ell$-ECTs of all the vertices $v\in V(G)$. Then $\{\ell\textnormal{-ECT}_1[{\mathcal{G}},x; v]\}_v$ provides all the (non-learnable) needed information to perform a sin

Figures (6)

  • Figure 1: Steps for computing the LEAP PE using $1$-hop neighborhoods. (1) The neighborhood of a node in a featured graph is selected. (2) Normalization of the neighborhood features. (3) Computation of the differentiable ECT. (4) Projection of the matrix representation of the ECT to get the PE vector.
  • Figure 2: Results for different PE strategies on the Alchemy and HIV datasets reporting the $R^2$ and AUROC respectively using the GCN architecture. Colors rank the PEs from best, second best to worst. LEAP with learnable direction significantly outperforms other methods on the Alchemy dataset while performing second best on the HIV dataset.
  • Figure 3: $R^2$ and AUROC results for different PE strategies on the Alchemy and HIV dataset using the GCN and GAT architectures. Best results are bold green, second best are green, and worst are red. LaPE achieves the best result for both architectures. For both architectures one of the two variants of our approach achieves the second best result.
  • Figure 4: Validation loss and accuracy per training epoch for the synthetic dataset for the baseline GCN, GAT, and LEAP. Our method achieves a perfect score in both metrics and convergence immediately. The shadows indicate one standard deviation over 5 runs and the dashed line means that model training finished earlier because of early stopping.
  • Figure 5: Validation accuracy per training epoch for the Letter High (left) and Letter Medium (right) datasets for different PE strategies using a GCN architecture. Our method achieves the best results and converges faster. The shadows around the curves indicate the standard deviation over $5$ runs and the dashed line means that training ended due to early stopping.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Definition 1
  • Theorem 1
  • Remark 1
  • Remark 2