Random Simplicial Complexes: Models and Phenomena

TL;DR

This paper reviews various models of random simplicial complexes, highlighting recent rigorous mathematical results, phenomena like phase transitions, and their applications as null models in real-world systems.

Contribution

It provides a focused overview of well-studied random simplicial complex models, emphasizing recent rigorous results and phenomena such as phase transitions and distributional limits.

Findings

Identification of key phase transitions in models

Distributional limits characterized for certain complexes

Emphasis on algebraic topology applications

Abstract

We review a collection of models of random simplicial complexes together with some of the most exciting phenomena related to them. We do not attempt to cover all existing models, but try to focus on those for which many important results have been recently established rigorously in mathematics, especially in the context of algebraic topology. In application to real-world systems, the reviewed models are typically used as null models, so that we take a statistical stance, emphasizing, where applicable, the entropic properties of the reviewed models. We also review a collection of phenomena and features observed in these models, and split the presented results into two classes: phase transitions and distributional limits. We conclude with an outline of interesting future research directions.