Demystifying Topological Message-Passing with Relational Structures: A Case Study on Oversquashing in Simplicial Message-Passing

TL;DR

This paper introduces a unifying axiomatic framework for topological message-passing that extends graph methods to higher-order structures, addressing oversquashing issues in topological deep learning.

Contribution

It provides a theoretical foundation connecting graph and topological message-passing, enabling analysis and mitigation of oversquashing in simplicial networks.

Findings

Framework extends graph algorithms to higher-order complexes

Theoretical insights into oversquashing phenomena

Empirical validation on simplicial networks

Abstract

Topological deep learning (TDL) has emerged as a powerful tool for modeling higher-order interactions in relational data. However, phenomena such as oversquashing in topological message-passing remain understudied and lack theoretical analysis. We propose a unifying axiomatic framework that bridges graph and topological message-passing by viewing simplicial and cellular complexes and their message-passing schemes through the lens of relational structures. This approach extends graph-theoretic results and algorithms to higher-order structures, facilitating the analysis and mitigation of oversquashing in topological message-passing networks. Through theoretical analysis and empirical studies on simplicial networks, we demonstrate the potential of this framework to advance TDL.