Homological percolation transitions in growing simplicial complexes

TL;DR

This paper investigates homological percolation transitions in growing simplicial complexes, revealing how higher-order topological features emerge and evolve in complex systems like social networks, through empirical data and a new model.

Contribution

It introduces a minimal model incorporating growth and preferential attachment to reproduce and analyze homological percolation transitions in simplicial complexes.

Findings

HPTs are characterized by first and second Betti numbers indicating cycles and cavities.

The model reproduces infinite order transition types with different critical exponents.

Delocalization of Betti numbers occurs due to merging and birth rates of simplexes.

Abstract

Simplicial complex (SC) representation is an elegant mathematical framework for representing the effect of complexes or groups with higher-order interactions in a variety of complex systems ranging from brain networks to social relationships. Here, we explore the homological percolation transitions (HPTs) of growing SCs using empirical datasets and a model proposed. The HPTs are determined by the first and second Betti numbers, which indicate the appearance of one- and two-dimensional macroscopic-scale homological cycles and cavities, respectively. A minimal SC model with two essential factors, namely, growth and preferential attachment, is proposed to model social coauthorship relationships. This model successfully reproduces the HPTs and determines the transition types as infinite order (the Berezinskii--Kosterlitz--Thouless type) with different critical exponents. In contrast to the Kahle localization observed in static random SCs, the first Betti number continues to increase even after the second Betti number appears. This delocalization is found to stem from the two aforementioned factors and arises when the merging rate of two-dimensional simplexes is less than the birth rate of isolated simplexes. Our results can provide topological insight into the maturing steps of complex networks such as social and biological networks.