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Do random initial degrees suppress concentration in preferential attachment graphs?

T. Makai, F. Polito, L. Sacerdote

TL;DR

The paper analyzes the PARID model with random initial degrees drawn from a power-law, showing a sharp threshold for concentration of the degree distribution: non-concentration for $α∈(1,2)$ and concentration for $α=2$ with explicit limits. The authors introduce a time-dependent truncation of the initial-degree distribution to handle infinite mean in the $α∈(1,2)$ regime and develop a framework of events and lemmas to quantify edge-insertion fluctuations that disrupt concentration. For the critical case $α=2$, they employ a slower-growing truncation and martingale concentration (Azuma–Hoeffding) to prove that the degree proportions converge to explicit $b_k$ given by $b_k= rac{2β(2)}{k(k+1)(k+2)}igl( frac{}{}igr)igl( extstyle rac{}{}igr) rac{}{} " $, confirming the conjecture in this regime. The results illuminate how heavy-tailed initial conditions influence concentration phenomena in preferential attachment graphs and establish rigorous behavior at the threshold exponent $α=2$.

Abstract

We consider the open problem concerning the possible lack of concentration of the degree distribution in preferential attachment graphs with random initial degree, when its distribution is characterized by extremely heavy tails of power-law type. We show that the addition of such a large number of edges causes a significant upset of the degree distribution, leading to its non-concentration. Furthermore, we show that the smallest value of the exponent for which the degree distribution exhibits concentration is 2.

Do random initial degrees suppress concentration in preferential attachment graphs?

TL;DR

The paper analyzes the PARID model with random initial degrees drawn from a power-law, showing a sharp threshold for concentration of the degree distribution: non-concentration for and concentration for with explicit limits. The authors introduce a time-dependent truncation of the initial-degree distribution to handle infinite mean in the regime and develop a framework of events and lemmas to quantify edge-insertion fluctuations that disrupt concentration. For the critical case , they employ a slower-growing truncation and martingale concentration (Azuma–Hoeffding) to prove that the degree proportions converge to explicit given by , confirming the conjecture in this regime. The results illuminate how heavy-tailed initial conditions influence concentration phenomena in preferential attachment graphs and establish rigorous behavior at the threshold exponent .

Abstract

We consider the open problem concerning the possible lack of concentration of the degree distribution in preferential attachment graphs with random initial degree, when its distribution is characterized by extremely heavy tails of power-law type. We show that the addition of such a large number of edges causes a significant upset of the degree distribution, leading to its non-concentration. Furthermore, we show that the smallest value of the exponent for which the degree distribution exhibits concentration is 2.
Paper Structure (9 sections, 18 theorems, 149 equations, 3 figures)

This paper contains 9 sections, 18 theorems, 149 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Proofs
  4. Proof of Theorem \ref{['thm:main']}
  5. Lemmas
  6. Proof of Theorem \ref{['thm:main']}
  7. Proof of Theorem \ref{['thm:main2']}
  8. Lemmas
  9. Proof of Theorem \ref{['thm:main2']}

Key Result

Theorem 2.1

Let $\alpha \in (1,2)$ and consider the PARID-model with parameters $\delta=0$ and $X$ which follows a power-law distribution with exponent $\alpha$. Then for any fixed $k\ge 1$ there exists no constant $a \in [0,1]$ such that $r_k(t)\stackrel{p}{\to}a$ as $t\to \infty$.

Figures (3)

  • Figure 1: Relationship between lemmas in the proof of Theorem \ref{['thm:main']}.
  • Figure 2: Relationship between lemmas in the proof of Theorem \ref{['thm:main2']}. For example the lemmas \ref{['trecinque']}, \ref{['lem:edgeconc']}, \ref{['lem:expR']}, \ref{['lem:marconc1']}, \ref{['lem:marconc2']} are those directly recalled by the proof of Theorem \ref{['thm:main2']}.
  • Figure 3: The process at the beginning of round (4,3), conditional on the $Z_j$-s.

Theorems & Definitions (34)

  • Theorem 2.1
  • Theorem 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 24 more