CCMamba: Topologically-Informed Selective State-Space Networks on Combinatorial Complexes for Higher-Order Graph Learning

TL;DR

CCMamba introduces a scalable, topologically-informed neural framework for higher-order graph learning that efficiently models long-range dependencies and surpasses existing methods in performance and robustness.

Contribution

It is the first to reformulate higher-order message passing as a linear, rank-aware sequence modeling problem using a Mamba-based neural architecture for combinatorial complexes.

Findings

Outperforms existing higher-order graph learning methods.

Demonstrates superior scalability and robustness in experiments.

Theoretically bounded by the 1-CCWL test.

Abstract

Topological deep learning has emerged as a powerful paradigm for modeling higher-order relational structures beyond pairwise interactions that standard graph neural networks fail to capture. While combinatorial complexes (CCs) offer a unified topological foundation for the higher-order graph learning, existing topological deep learning methods rely heavily on local message passing and attention mechanisms. These suffer from quadratic complexity and local neighborhood constraints, limiting their scalability and capacity for rank-aware, long-range dependency modeling. To overcome these challenges, we propose Combinatorial Complex Mamba (CCMamba), the first unified Mamba-based neural framework for learning on combinatorial complexes. CCMamba reformulates higher-order message passing as a selective state-space modeling problem by linearizing multi-rank incidence relations into structured, rank-aware sequences. This architecture enables adaptive, directional, and long-range information propagation in linear time bypassing the scalability bottlenecks of self-attention. Theoretically, we further establish that the expressive power of CCMamba is upper-bounded by the 1-dimensional combinatorial complex Weisfeiler-Lehman (1-CCWL) test. Extensive experiments across graph, hypergraph, and simplicial benchmarks demonstrate that CCMamba consistently outperforms existing methods while exhibiting superior scalability and remarkable robustness against over-smoothing in deep architectures.