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

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 messagepassing networks. Through theoretical analysis and empirical studies on simplicial networks, we demonstrate the potential of this framework to advance TDL. * Equal contribution. † Equal contribution. 1. Boundary adjacency: B(σ) = τ : τ ≺ σ; 2. Co-boundary adjacency: C(σ) = τ : σ ≺ τ ; 3. Lower adjacency: N ↓ (σ) = τ : ∃δ such that δ ≺ τ and δ ≺ σ; 4. Upper adjacency: N ↑ (σ) = τ : ∃δ such that τ ≺ δ and σ ≺ δ. In Figure 1 , we illustrate an example of a simplicial complex and its adjacency relations. We now, following Bodnar et al. (2021b, Section 4), review a general scheme for message passing on simplicial complexes. In Appendix A, we provide references for topological message passing architectures that fit this scheme. We refer readers to Appendix F.5 for specific instantiations of this scheme in our graph and topological message passing models.