CCMamba: Selective State-Space Models for Higher-Order Graph Learning on Combinatorial Complexes

TL;DR

This work tackles the challenge of learning on higher-order relational structures by proposing CCMamba, a unified Mamba-based framework that treats combinatorial complex message passing as rank-aware selective state-space modeling to achieve linear-time propagation. By encoding incidence relations across ranks into structured sequences processed by selective SSMs, CCMamba enables directional and long-range information flow across nodes, edges, and higher-order cells, generalizing to graphs, hypergraphs, simplicial, and cellular complexes. The authors establish a theoretical expressivity bound: CCMamba's discriminative power is upper-bounded by the $1$-CCWL test, and they show that this bound can be attained with appropriate, injective readouts. Empirically, CCMamba variants outperform attention-based and other higher-order baselines across graph and node classification tasks, while offering substantial efficiency gains and robustness to depth, highlighting its scalability and practical impact for topological deep learning on complex structures.

Abstract

Topological deep learning has emerged for modeling higher-order relational structures beyond pairwise interactions that standard graph neural networks fail to capture. Although combinatorial complexes offer a unified topological framework, most existing topological deep learning methods rely on local message passing via attention mechanisms, which incur quadratic complexity and remain low-dimensional, limiting scalability and rank-aware information aggregation in higher-order complexes.We propose Combinatorial Complex Mamba (CCMamba), the first unified mamba-based neural framework for learning on combinatorial complexes. CCMamba reformulates message passing as a selective state-space modeling problem by organizing multi-rank incidence relations into structured sequences processed by rank-aware state-space models. This enables adaptive, directional, and long range information propagation in linear time without self attention. We further establish the theoretical analysis that the expressive power upper-bound of CCMamba message passing is the 1-Weisfeiler-Lehman test. Experiments on graph, hypergraph, and simplicial benchmarks demonstrate that CCMamba consistently outperforms existing methods while exhibiting improved scalability and robustness to depth.

Figures (3)

• Figure 1: Illustration of combinatorial complex
• Figure 2: The framework of our proposed Combinatorial Complex Mamba method.
• Figure 3: Comparison of GPU Memory (MB) and Time Cost(s).

Theorems & Definitions (16)

• Definition 1: Combinatorial Complex
• Definition 2: Lifting Operation
• Definition 3: (Labeled Combinatorial Complex)
• Definition 4: Combinatorial Complex Weisfeiler Leman (CCWL)
• Proposition 1: (1-CCWL)
• Lemma 1
• Proposition 2
• Theorem 1
• Proposition 3: (1-CCWL)
• proof : Proof for Proposition \ref{['proposition01']}