Degree counts in random simplicial complexes of the preferential attachment type

TL;DR

This paper generalizes the preferential attachment model to higher-dimensional simplicial complexes, establishing laws of large numbers, power-law degree distributions, and multivariate limit theorems for degree counts across dimensions.

Contribution

It introduces a novel preferential attachment model for simplicial complexes and derives asymptotic degree distribution laws and multivariate convergence results.

Findings

Degree counts follow a mixture of negative binomial distributions.

Degree distributions exhibit power-law behavior.

Multivariate limits relate to independent linear birth processes.

Abstract

We extend the classical preferential attachment random graph model to random simplicial complexes. At each stage of the model, we choose one of the existing $k$-simplices with probability proportional to its $k$-degree. The chosen $k$-simplex then forms a $(k+1)$-simplex with a newly arriving vertex. We establish a strong law of large numbers for the degree counts across multiple dimensions. The limiting probability mass function is expressed as a mixture of mass functions of different types of negative binomial random variables. This limiting distribution has power-law characteristics and we explore the limiting extremal dependence of the degree counts across different dimensions in the framework of multivariate regular variation. Finally, we prove multivariate weak convergence, under appropriate normalization, of degree counts in different dimensions, of ordered $k$-simplices. The resulting weak limit can be represented as a function of independent linear birth processes with immigration.