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Degree counts in random simplicial complexes of the preferential attachment type

Takashi Owada, Gennady Samorodnitsky

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.

Degree counts in random simplicial complexes of the preferential attachment type

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 -simplices with probability proportional to its -degree. The chosen -simplex then forms a -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 -simplices. The resulting weak limit can be represented as a function of independent linear birth processes with immigration.

Paper Structure

This paper contains 12 sections, 14 theorems, 145 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preferential attachment model
  3. Setup
  4. Multivariate regular variation of degree distributions
  5. Degrees of fixed $k$-simplices
  6. Proofs
  7. Proof of Theorem \ref{['t:SLLN']}
  8. Proof of Proposition \ref{['p:representation']}
  9. Proof of Theorem \ref{['t:regular.variation']}
  10. Proof of Proposition \ref{['p:degree.and.BI']}
  11. Proof of Theorem \ref{['t:joint.weak.conv']}
  12. Appendix

Key Result

Theorem 2.1

For every $1\le i_k<\dots <i_0$, the sequence of random pmfs defined in e:empirical.pn satisfies where $(p_{i_0,\dots,i_k})_{1\le i_k < \dots <i_0}$ is a pmf, defined recursively by where

Figures (1)

  • Figure 1: Assume that $k=2$ and $\delta=0$. (a) At time $0$, we start with the $3$-simplex on a vertex set $\{ 1,2,3,4 \}$. (b) At time $1$, each of the $2$-simplices $\{ 1,2,3 \}, \{ 1,2,4 \}, \{ 1,3,4 \}$, and $\{ 2,3,4 \}$ is chosen with equal probability, i.e., $1/4$ each. Suppose $\{ 1,3,4 \}$ is selected at this stage. (c) At time $2$, there are initially seven $2$-simplices: $\{ 1,2,3 \}, \{ 1,2,4 \}, \{ 1,3,4 \}, \{ 2,3,4 \}, \{ 1,3,5 \}, \{ 1,4,5 \}$, and $\{ 3,4,5 \}$. Here, the $2$-degree of $\{ 1,3,4 \}$ is $2$, while all other six $2$-simplices have a $2$-degree $1$. Therefore, the $2$-simplices $\{ 1,3,4 \}$ is chosen with probability $1/4$, whereas all other six $2$-simplices are selected with probability $1/8$.

Theorems & Definitions (35)

  • Definition 1.1
  • Theorem 2.1
  • Remark 2.2
  • Proposition 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • Proposition 3.1
  • Theorem 3.2
  • ...and 25 more