Random Abstract Cell Complexes

2024-06-04

arXiv.org

AI Summary

Scalable random models for high-dimensional topological data analysis (TDA) remain scarce, particularly for abstract cell complexes (CCs). Method: We introduce the first Erdős–Rényi–style random CC model, constructing complexes layerwise by dimension via probabilistic cell addition, with a focus on overcoming sampling bottlenecks in the 2D case. Our approach features a novel boundary-length-constrained 2-cell sampling mechanism and a fast, enumeration-free estimator for the number of simple cycles in a graph. By integrating probabilistic graphical modeling, combinatorial approximation, and randomized algorithms, we achieve tunable-distribution sampling of 2D random CCs. Contribution/Results: We release py-raccoon, an open-source toolkit implementing the model. Experiments validate its effectiveness as a null model in TDA and as a graph augmentation tool for topological feature enhancement, demonstrating both theoretical soundness and practical utility.

Technology Category

Machine Learning: Probabilistic Circuits and Graphical ModelsReasoning under Uncertainty: Graphical ModelsKnowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsWeb Mining and Content Analysis: Models for Web evolutionSemantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We define a model for random (abstract) cell complexes (CCs), similiar to the well-known ErdH{o}s-R'enyi model for graphs and its extensions for simplicial complexes. To build a random cell complex, we first draw from an ErdH{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. Since there is a large variance in the number of simple cycles on graphs drawn from the same configuration of ER, we also provide an efficient method to approximate that number, which is of independent interest. Moreover, it enables us to specify the expected number of 2-cells of each boundary length we want to sample. 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. Both the sampling and cycle count estimation algorithms are available in the package `py-raccoon` on the Python Packaging Index.

Problem

• Defining a random model for abstract cell complexes similar to Erdős-Rényi graphs
• Developing sampling algorithms for two-dimensional cell complexes with probability control
• Approximating cycle-related graph statistics using importance sampling techniques

Innovation

• Random cell complex model extends Erdős-Rényi graphs
• Spanning-tree-based algorithm samples cycles efficiently
• Approximates cycle statistics using importance sampling technique