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

TL;DR

This paper introduces simplified, practical definitions of cell complexes rooted in algebra, making them more accessible for applications in signal processing and network science, especially for lower-dimensional cases.

Contribution

It provides an accessible, algebra-based definition of cell complexes and bridges the gap between abstract topology and practical signal processing applications.

Findings

Simplified definitions of cell complexes for practical use.

Equivalence of algebraic and topological notions in most applications.

Enhanced understanding of 2-dimensional and lower cell complexes.

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.