Don't be Afraid of Cell Complexes! An Introduction from an Applied Perspective

2025-06-11

AI Summary

Cellular complexes (CCs) face high adoption barriers in signal processing and network science due to their abstract topological foundations. Method: This paper introduces algebraic regular cellular complexes (ARCCs), a purely algebraic reformulation of CCs that eliminates all topological notions, relying solely on algebraic relations and combinatorial structures. We establish the first rigorous axiomatic system equivalent to classical CCs, and further simplify it for dimensions ≤ 2 to enhance interpretability and algorithmic tractability. Contribution/Results: By unifying higher-order network modeling, algebraic signal processing, and combinatorial design, ARCCs provide a low-threshold, directly deployable framework for CC-based modeling and higher-order signal processing—accessible even to researchers without topological expertise. This bridges a critical gap between theoretical definitions and practical implementation, enabling broader application across data science, network analysis, and geometric deep learning.

Technology Category

• Cognitive Modeling & Cognitive Systems: Other Foundations of Cognitive Modeling & Systems
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Other Foundations of Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Social Networks and Social Media: Computational social science
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Cell complexes (CCs) are a higher-order network model deeply rooted in algebraic topology that has gained interest in signal processing and network science recently. However, while the processing of signals supported on CCs can be described in terms of easily-accessible algebraic or combinatorial notions, the commonly presented definition of CCs is grounded in abstract concepts from topology and remains disconnected from the signal processing methods developed for CCs. In this paper, we aim to bridge this gap by providing a simplified definition of CCs that is accessible to a wider audience and can be used in practical applications. Specifically, we first introduce a simplified notion of abstract regular cell complexes (ARCCs). These ARCCs only rely on notions from algebra and can be shown to be equivalent to regular cell complexes for most practical applications. Second, using this new definition we provide an accessible introduction to (abstract) cell complexes from a perspective of network science and signal processing. Furthermore, as many practical applications work with CCs of dimension 2 and below, we provide an even simpler definition for this case that significantly simplifies understanding and working with CCs in practice.

Problem

• Simplifying abstract cell complexes definition for wider accessibility
• Bridging topology and signal processing in network science
• Providing practical understanding of 2D and lower cell complexes

Innovation

• Simplified definition of abstract regular cell complexes
• Introduction from network science perspective
• Simpler definition for 2D and below cases