PlanE: Representation Learning over Planar Graphs

TL;DR

This paper introduces PlanE, a new graph neural network framework designed to learn complete invariants for planar graphs, leveraging classical algorithms to improve isomorphism testing and achieve state-of-the-art results.

Contribution

PlanE is the first architecture to efficiently learn complete invariants for planar graphs, inspired by classical isomorphism algorithms, and scalable in practice.

Findings

Achieves state-of-the-art performance on planar graph benchmarks.

Successfully learns complete invariants for planar graphs.

Demonstrates scalability and effectiveness of the proposed architectures.

Abstract

Graph neural networks are prominent models for representation learning over graphs, where the idea is to iteratively compute representations of nodes of an input graph through a series of transformations in such a way that the learned graph function is isomorphism invariant on graphs, which makes the learned representations graph invariants. On the other hand, it is well-known that graph invariants learned by these class of models are incomplete: there are pairs of non-isomorphic graphs which cannot be distinguished by standard graph neural networks. This is unsurprising given the computational difficulty of graph isomorphism testing on general graphs, but the situation begs to differ for special graph classes, for which efficient graph isomorphism testing algorithms are known, such as planar graphs. The goal of this work is to design architectures for efficiently learning complete invariants of planar graphs. Inspired by the classical planar graph isomorphism algorithm of Hopcroft and Tarjan, we propose PlanE as a framework for planar representation learning. PlanE includes architectures which can learn complete invariants over planar graphs while remaining practically scalable. We empirically validate the strong performance of the resulting model architectures on well-known planar graph benchmarks, achieving multiple state-of-the-art results.