Homological percolation transitions in growing simplicial complexes

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 model studies. 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 an infinite-order Berezinskii-Kosterlitz-Thouless type but 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 a topological insight into the maturing steps of complex networks such as social and biological networks.

Automated summary

The simplicial complex representation is a mathematical framework for describing the high-order interaction effects of complex groups in various systems. The homological percolation transitions can be determined by the first and second Betti numbers. A minimal SC model with growth and preferential attachment factors successfully reproduces the HPTs in social coauthorship relationships.