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Asymptotic normality of degree counts in a general preferential attachment model

Simone Baldassarri, Gianmarco Bet

Abstract

We consider the preferential attachment model. This is a growing random graph such that at each step a new vertex is added and forms $m$ connections. The neighbors of the new vertex are chosen at random with probability proportional to their degree. It is well known that the proportion of nodes with a given degree at step $n$ converges to a constant as $n\rightarrow\infty$. The goal of this paper is to investigate the asymptotic distribution of the fluctuations around this limiting value. We prove a central limit theorem for the joint distribution of all degree counts. In particular, we give an explicit expression for the asymptotic covariance. This expression is rather complex, so we compute it numerically for various parameter choices. We also use numerical simulations to argue that the convergence is quite fast. The proof relies on the careful construction of an appropriate martingale.

Asymptotic normality of degree counts in a general preferential attachment model

Abstract

We consider the preferential attachment model. This is a growing random graph such that at each step a new vertex is added and forms connections. The neighbors of the new vertex are chosen at random with probability proportional to their degree. It is well known that the proportion of nodes with a given degree at step converges to a constant as . The goal of this paper is to investigate the asymptotic distribution of the fluctuations around this limiting value. We prove a central limit theorem for the joint distribution of all degree counts. In particular, we give an explicit expression for the asymptotic covariance. This expression is rather complex, so we compute it numerically for various parameter choices. We also use numerical simulations to argue that the convergence is quite fast. The proof relies on the careful construction of an appropriate martingale.

Paper Structure

This paper contains 6 sections, 2 theorems, 87 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Model and results
  3. Proof of the main result
  4. First condition of Theorem \ref{['tlc']}
  5. Second condition of Theorem \ref{['tlc']}
  6. The limit process for degree counts

Key Result

Theorem 2.5

As $s\rightarrow\infty$, where $(Z_k, \ k=m,m+1,...)$ is a mean zero Gaussian process with covariance function $R_Z$ given by for $r,l\geq m$. Here $b_j^{(k)}$ are given by

Figures (4)

  • Figure 1: $R_Z(r,r)$ for $m=1$ and ${\delta}=-0.5$, $0$, $1$, $3$, $5$ on the left-hand side and $R_Z(r,r)$ for $m=2$ and ${\delta}=-1$, $0$, $2$, $6$, $10$ on the right-hand side. To highlight the different behaviour of the various variance functions, we use the logarithmic axis on the $y$-axis.
  • Figure 2: $R_Z(r,r)$ for ${\delta}=0$ and $m=1,2,3$. To highlight the different behaviour of the various variance functions, we use the logarithmic axis on the $y$-axis.
  • Figure 3: $R_Z(r,5)$ for ${\delta}=0$ and $m=1$, $2$, $3$.
  • Figure 4: Left-hand side: $R_Z(r,r)$ for ${\delta}=1$ and $m=1$ ($t=\infty$) compared with the numerical simulations stopped at time $t=100,1000,5000$. Each empirical curve was obtained by taking the average of $N=10000$ simulations. To highlight the different behaviour of the various variance functions, we use the logarithmic axis on the $y$-axis. Right-hand side: $R_Z(r,5)$ for $m=2$ and ${\delta}=0$ compared with the numerical simulations stopped at time $t=100,1000,5000$. Each empirical curve was obtained by taking the average of $N=10000$ simulations.

Theorems & Definitions (4)

  • Theorem 2.5
  • Remark 2.9
  • Theorem 3.1
  • Remark 3.14