Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Logarithmic typical distances in preferential attachment models

Remco van der Hofstad, Haodong Zhu

TL;DR

This work resolves the logarithmic scaling of typical distances in sparse preferential attachment networks with Out-degree $m\ge 2$ and positive fitness $\delta$, proving dist$(o_1,o_2)\sim \log_{\nu} n$ in probability, where $\nu$ is the local-limit growth parameter. The authors develop a robust path-counting framework built on a Pólya urn representation, relate local structure to a Pólya point tree, and analyze a truncated offspring operator via a novel probabilistic approach to its spectral radius. The proof combines first- and second-moment methods with careful truncation and a detailed decomposition of path dependencies, ultimately showing the truncated spectral radius converges to $\nu$ and delivering the sharp bound for typical distances. Extensions to PAM variations are discussed through a collapsing/compatibility framework, reinforcing a universal logarithmic-distance behavior in this class of models. The results advance understanding of distance scales in PAMs and connect local-limit theory to global graph metrics with precise asymptotics.

Abstract

We prove that the typical distances in a preferential attachment model with out-degree $m\geq 2$ and strictly positive fitness parameter are close to $\log_ν{n}$, where $ν$ is the exponential growth parameter of the local limit of the preferential attachment model. The proof relies on a path-counting technique, the first- and second-moment methods, as well as a novel proof of the convergence of the spectral radius of the offspring operator under a certain truncation.

Logarithmic typical distances in preferential attachment models

TL;DR

This work resolves the logarithmic scaling of typical distances in sparse preferential attachment networks with Out-degree and positive fitness , proving dist in probability, where is the local-limit growth parameter. The authors develop a robust path-counting framework built on a Pólya urn representation, relate local structure to a Pólya point tree, and analyze a truncated offspring operator via a novel probabilistic approach to its spectral radius. The proof combines first- and second-moment methods with careful truncation and a detailed decomposition of path dependencies, ultimately showing the truncated spectral radius converges to and delivering the sharp bound for typical distances. Extensions to PAM variations are discussed through a collapsing/compatibility framework, reinforcing a universal logarithmic-distance behavior in this class of models. The results advance understanding of distance scales in PAMs and connect local-limit theory to global graph metrics with precise asymptotics.

Abstract

We prove that the typical distances in a preferential attachment model with out-degree and strictly positive fitness parameter are close to , where is the exponential growth parameter of the local limit of the preferential attachment model. The proof relies on a path-counting technique, the first- and second-moment methods, as well as a novel proof of the convergence of the spectral radius of the offspring operator under a certain truncation.

Paper Structure

This paper contains 56 sections, 23 theorems, 303 equations, 3 figures.

Table of Contents

  1. Introduction and main result
  2. Introduction
  3. Common properties of real-world networks
  4. Universality of the typical distance and the exponential growth parameter $\nu$
  5. Model definition
  6. Construction of ${\rm PA}_{n}^{ (m,\delta)}(d)$
  7. Connectivity of ${\rm PA}_{n}^{ (m,\delta)}(d)$
  8. Main result
  9. Discussion
  10. Challenges in analyzing the typical distances for PAMs
  11. Open problem
  12. Organization
  13. Asymptotic symbols and abbreviations
  14. Proof overview
  15. The Pólya urn representation of ${\rm PA}_{n}$
  16. ...and 41 more sections

Key Result

Theorem 1.1

Fix $m\geq 2$ and $\delta>0$, and consider ${\rm PA}_{n}^{ (m,\delta)}(d)$. As $n\rightarrow \infty$, with $o_1,o_2$ uniformly distributed in $[n]$, where ${\rm dist}_{ {\rm PA}_{n}^{ (m,\delta)}(d)}(o_1, o_2)$ denotes the length of the shortest path between $o_1$ and $o_2$ in ${\rm PA}_{n}^{ (m,\delta)}(d)$, and

Figures (3)

  • Figure 1: A sample of the path decomposition with decomposition size $u=3$
  • Figure 2: Two cases for the path decomposition of $\vec{\rho}^{ e}$ arise when decomposition size $u=1$ and $\rho(1)_0\in\vec{\pi}$, where this categorization depends on whether $\rho(1)_\ell$ is in $\vec{\pi}$. The case where $\rho(1)_0\not\in\vec{\pi}$ and $\rho(1)_\ell\in\vec{\pi}$ is simply the reverse of the first graph.
  • Figure 3: A sample of the decomposition

Theorems & Definitions (60)

  • Theorem 1.1: Convergence of the typical distances
  • Remark 2.1: Pólya urn graph allowing self-loops
  • Definition 2.2: Edge
  • Definition 2.3: Edge set
  • Definition 2.4: Path and edge-labeled paths
  • Proposition 2.5: Lower bound on the typical distances
  • Definition 2.6: Good neighborhood pair
  • Definition 2.7: Good weights
  • Definition 2.8: Good event $\mathcal{E}$
  • Lemma 2.9: Good events have high probability
  • ...and 50 more