Random Abstract Cell Complexes

TL;DR

This paper introduces a probabilistic model for random abstract cell complexes based on Erdős-Rényi graphs, along with an efficient sampling algorithm for 2-dimensional complexes, enabling analysis of their properties and applications as null models.

Contribution

It presents a novel model for random cell complexes, an approximate sampling algorithm for 2D complexes, and methods for cycle analysis, expanding tools for topological data analysis.

Findings

Developed an approximate sampling algorithm for 2D cell complexes.

Enabled estimation of cycle counts and probabilities in random complexes.

Provided practical tools and applications for null models and graph liftings.

Abstract

We define a model for random (abstract) cell complexes (CCs), similiar to the well-known Erd\H{o}s-R\'enyi model for graphs and its extensions for simplicial complexes. To build a random cell complex, we first draw from an Erd\H{o}s-R\'enyi graph, and consecutively augment the graph with cells for each dimension with a specified probability. As the number of possible cells increases combinatorially -- e.g., 2-cells can be represented as cycles, or permutations -- we derive an approximate sampling algorithm for this model limited to two-dimensional abstract cell complexes. As a basis for this algorithm, we first introduce a spanning-tree-based method that samples simple cycles and allows the efficient approximation of various properties, most notably the probability of occurence of a given cycle. This approximation is of independent interest as it enables the approximation of a wide variety of cycle-related graph statistics using importance sampling. We use this to approximate the number of cycles of a given length on a graph, allowing us to calculate the sampling probability to arrive at a desired expected number of sampled 2-cells. The probability approximation also trivially leads to a sampling algorithm for $2$-cells with a desired sampling probability. We provide some initial analysis into the properties of random CCs drawn from this model. We further showcase practical applications for our random CCs as null models, and in the context of (random) liftings of graphs to cell complexes. The cycle sampling, cycle count estimation, and combined cell sampling algorithms are available in the package `py-raccoon` on the Python Packaging Index.