Ccmamba enables scalable higher-order graph learning with combinatorial complex numbers with quadratic complexity

Researchers are addressing the limitations of current graph neural networks by exploring ways to better model complex relationships beyond simple pairwise interactions. Jiawen Chen, Qi Shao, and Mintong Zhou from the Department of Mathematics at Southeast University, along with Duxin Chen and Wenwu Yu, present a novel approach in their paper “CCMamba: Selective state-space model for higher-order graph learning with combinatorial complex numbers.” They introduce Combinatorial Complex Mamba (CCMamba), a breakthrough framework that leverages state-space models to efficiently learn from combinatorial complexes, which are unified topological structures, overcoming the second-order complexity and limited dimensionality of existing attention-based methods. This innovation not only improves the scalability and robustness of higher-order complexes, but also demonstrates expressive power comparable to the powerful 1-Weisfeiler-Lehman test, representing a major advance in topological machine learning.

CCMamba learns from and generalizes from complex relational structures.

This breakthrough addresses the limitations of existing methods that struggle to model higher-order relational structures beyond simple pairwise interactions, a common challenge when analyzing complex systems. CCMamba’s core innovation lies in its ability to efficiently handle complex topologies of graphs, hypergraphs, and combinatorial complexes that generalize higher-order complexes into unified structures. By treating topological relationships as structured sequences, the researchers unlocked the potential of state-space models to achieve global, context-aware propagation, overcoming the locality constraints of previous topological deep learning approaches. Furthermore, this work established the theoretical foundation of CCMamba’s expressive power and proved that its message passing capabilities reach the upper bound of the 1-Weisfeiler, Lehman test, an important benchmark for graph learning.
This theoretical analysis solidifies the potential of the framework to capture complex relationships within the data. This research opens new avenues for applying topological deep learning to real-world problems such as biomolecular interaction networks, traffic dynamics, and image analysis. This study establishes a new paradigm for high-order graph learning, going beyond the limitations of low-dimensional message passing and quadratic complexity. CCMamba provides a scalable and expressive framework for capturing complex relationship patterns in complex data by integrating selective state-space models.

Measurements confirm that CCMamba achieves this efficiency without compromising expressiveness. Theoretical analysis has established that its upper bound on message passing ability is equivalent to the 1-Weisfeiler-Lehman test. The results demonstrate CCMamba’s ability to process complex data with significantly reduced computational demands. Specifically, this framework linearizes neighborhood sequences and models long-range propagation, providing significant improvements over traditional methods. In this work, we elaborately define lifting operations to transform graph data into higher-order combinatorial complexes, allowing us to express complex relationships beyond simple pairwise interactions.
This allows for a more nuanced understanding of the underlying structure of the data. Tests prove that CCMamba’s rank-structured selective state-space mechanism effectively captures intra- and inter-cell dependencies of different topological ranks. The researchers defined four main neighborhood types: node-edge, edge-node, edge-face, and face-edge to control message propagation between dimensions and ensure comprehensive information flow. The data show that the framework accurately models these relationships and achieves good performance on node and graph classification tasks. The expressive power of CC-based Mamba message passing is limited by a one-dimensional combinatorial complex Weisfeiler, Lehman test, confirming its theoretical capabilities.

Furthermore, this study establishes a formal definition of a combinatorial complex as a triple (S, C, rk). Here S is a finite set of vertices, C is a set of cells, and rk is a rank function that preserves order. This rigorous mathematical foundation underpins the framework’s ability to generalize across a variety of data types, including graphs, hypergraphs, and simple complexes. This breakthrough provides a scalable and robust solution for learning complex data, opening new avenues for applications in fields as diverse as materials science, social network analysis, and drug discovery.

CCMamba learns complex structures efficiently and quickly

In this study, message passing, a key process in neural networks, is reformulated as a selective state modeling problem to enable efficient information processing. The importance of this work lies in its ability to generalize across a variety of complex data types, including graphs, hypergraphs, simple complexes, and cellular complexes, while maintaining expressive power comparable to the 1-Weisfeiler-Lehman test, which is a measure of a model’s ability to distinguish between different network structures. The authors acknowledge that the current implementation is limited by its focus on node classification tasks, and that further research is needed to investigate performance on other tasks. Future research directions include extending CCMamba to handle dynamic combinatorial complexes and considering applications to more complex real-world datasets, potentially unlocking new capabilities in areas such as materials discovery and drug discovery.