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Universality of the local limit of preferential attachment models

Alessandro Garavaglia, Rajat Subhra Hazra, Remco van der Hofstad, Rounak Ray

TL;DR

The paper develops a universal local limit for a broad class of preferential attachment models with i.i.d. out-degrees by proving convergence to a Random Pólya Point Tree (RPPT). It achieves this via a two-pronged approach: (i) representing PAMs as collapsed Pólya urn graphs to obtain a conditional independence structure, and (ii) deriving explicit neighbourhood densities and proving density convergence to RPPT. The results hold for multiple PAM variants, including those with negative fitness and infinite-variance degrees, establishing universality of the local limit and providing detailed consequences for degree distributions and neighbourhood structure. The work also extends the literature by providing density convergence (not just local weak convergence) and by coupling models without fetching restrictions, tying together several PAMs under a single RPPT framework and opening directions for percolation and distance analyses.

Abstract

We study preferential attachment models where vertices enter the network with i.i.d. random numbers of edges that we call the out-degree. We identify the local limit of such models, substantially extending the work of Berger et al.(2014). The degree distribution of this limiting random graph, which we call the random Pólya point tree, has a surprising size-biasing phenomenon. Many of the existing preferential attachment models can be viewed as special cases of our preferential attachment model with i.i.d. out-degrees. Additionally, our models incorporate negative values of the preferential attachment fitness parameter, which allows us to consider preferential attachment models with infinite-variance degrees. Our proof of local convergence consists of two main steps: a Pólya urn description of our graphs, and an explicit identification of the neighbourhoods in them. We provide a novel and explicit proof to establish a coupling between the preferential attachment model and the Pólya urn graph. Our result proves a density convergence result, for fixed ages of vertices in the local limit.

Universality of the local limit of preferential attachment models

TL;DR

The paper develops a universal local limit for a broad class of preferential attachment models with i.i.d. out-degrees by proving convergence to a Random Pólya Point Tree (RPPT). It achieves this via a two-pronged approach: (i) representing PAMs as collapsed Pólya urn graphs to obtain a conditional independence structure, and (ii) deriving explicit neighbourhood densities and proving density convergence to RPPT. The results hold for multiple PAM variants, including those with negative fitness and infinite-variance degrees, establishing universality of the local limit and providing detailed consequences for degree distributions and neighbourhood structure. The work also extends the literature by providing density convergence (not just local weak convergence) and by coupling models without fetching restrictions, tying together several PAMs under a single RPPT framework and opening directions for percolation and distance analyses.

Abstract

We study preferential attachment models where vertices enter the network with i.i.d. random numbers of edges that we call the out-degree. We identify the local limit of such models, substantially extending the work of Berger et al.(2014). The degree distribution of this limiting random graph, which we call the random Pólya point tree, has a surprising size-biasing phenomenon. Many of the existing preferential attachment models can be viewed as special cases of our preferential attachment model with i.i.d. out-degrees. Additionally, our models incorporate negative values of the preferential attachment fitness parameter, which allows us to consider preferential attachment models with infinite-variance degrees. Our proof of local convergence consists of two main steps: a Pólya urn description of our graphs, and an explicit identification of the neighbourhoods in them. We provide a novel and explicit proof to establish a coupling between the preferential attachment model and the Pólya urn graph. Our result proves a density convergence result, for fixed ages of vertices in the local limit.
Paper Structure (59 sections, 36 theorems, 306 equations, 2 figures)

This paper contains 59 sections, 36 theorems, 306 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Real-world networks and preferential attachment models
  3. Main results and innovation of this paper
  4. The models
  5. Model (A)
  6. Model (B)
  7. Model (D)
  8. Model (E)
  9. Model (F)
  10. The space of rooted vertex-marked graphs and marked local convergence
  11. Definition of random Pólya point tree
  12. Random Pólya Point Tree (RPPT)
  13. Descriptions of the distributions and parameters used
  14. Feature of the vertices of RPPT
  15. Construction of the RPPT
  16. ...and 44 more sections

Key Result

Theorem 1.5

Let $M$ be an $\mathbb{N}$-valued random variable with finite $p$th moment for some $p>1$ and $\delta>-\operatorname{inf~supp}(M)$. Then models $(A), (B), (D),(E)$ and $(F)$ of the preferential attachment converge vertex-marked locally in probability to the random Pólya point tree with parameters $M

Figures (2)

  • Figure 1: Creation of ${ {\sf O}}$-labelled children of $(a_{\omega},t_{\omega})$
  • Figure 2: Creation of ${\sf Y}$-labelled children of $(a_{\omega},t_{\omega})$

Theorems & Definitions (79)

  • Remark 1.1: Difference between Models (A) and (B)
  • Remark 1.2: The missing model (c)
  • Definition 1.3: Vertex marked local convergence
  • Definition 1.4: Ulam-Harris set and its ordering
  • Theorem 1.5: Local convergence theorem for PA models
  • Corollary 1.6: Asymptotic degree distribution for older and younger neighbours
  • Definition 2.1: Pólya Urn Graphs
  • Remark 2.2: Initial graph is preserved in collapsing
  • Definition 2.3: Collapsed Pólya Urn graph
  • Remark 2.4: Self-loops for $\mathrm{CPU}_n^{ (\mathrm{NSL})}$
  • ...and 69 more