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Local weak convergence and its applications

Sayan Banerjee, Shankar Bhamidi, Jianan Shen, Seth Parker Young

TL;DR

The main goal of this paper is to give an overview of local weak convergence, which has emerged as a major technique for understanding large network asymptotics for a wide array of functionals and models.

Abstract

Motivated in part by understanding average case analysis of fundamental algorithms in computer science, and in part by the wide array of network data available over the last decade, a variety of random graph models, with corresponding processes on these objects, have been proposed over the last few years. The main goal of this paper is to give an overview of local weak convergence, which has emerged as a major technique for understanding large network asymptotics for a wide array of functionals and models. As opposed to a survey, the main goal is to try to explain some of the major concepts and their use to junior researchers in the field and indicate potential resources for further reading.

Local weak convergence and its applications

TL;DR

The main goal of this paper is to give an overview of local weak convergence, which has emerged as a major technique for understanding large network asymptotics for a wide array of functionals and models.

Abstract

Motivated in part by understanding average case analysis of fundamental algorithms in computer science, and in part by the wide array of network data available over the last decade, a variety of random graph models, with corresponding processes on these objects, have been proposed over the last few years. The main goal of this paper is to give an overview of local weak convergence, which has emerged as a major technique for understanding large network asymptotics for a wide array of functionals and models. As opposed to a survey, the main goal is to try to explain some of the major concepts and their use to junior researchers in the field and indicate potential resources for further reading.
Paper Structure (28 sections, 13 theorems, 41 equations, 14 figures)

This paper contains 28 sections, 13 theorems, 41 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Motivating examples and questions
  3. Dynamic social network models
  4. Google's PageRank
  5. Gibbs distribution on graphs
  6. Spectral distribution of adjacency matrices
  7. Probabilistic combinatorial optimization
  8. Definitions and notation
  9. Weak convergence: General theory
  10. Local weak convergence (LWC): Intuition and general framework
  11. Putting the "local" in LWC: lifting convergence to the space of probability measures
  12. Special case: Local weak convergence of trees
  13. Convergence on the space of trees
  14. Extensions
  15. Examples
  16. ...and 13 more sections

Key Result

Theorem 3.5

Let $\left\{\mathcal{G}_n:n\geqslant 1\right\}$ be a sequence of (potentially random) elements in $\mathbb{G}_{<\infty}$ and let $\mathop{\mathrm{\mathbb{P}}}\nolimits_\infty$ be a probability measure on $(\mathbb{G}_{\star}, \mathcal{B}(\mathbb{G}_{\star}))$ and let $\mathcal{G}_{\infty, \star}\sim

Figures (14)

  • Figure 2.1: A simulation with two types ($\mathcal{S} = \{$red, blue$\}$) with $n=30,000$ and $\kappa(a,a) = .75$, $\kappa(a,a^\prime) = .25$ for $a\neq a^\prime$ and $\pi(red) = .35$ in the linear $\gamma \equiv 1$ case.
  • Figure 3.1: Generated via imgflip.com
  • Figure 3.2: Fringe decomposition around vertex $v$ of a finite tree rooted at $\rho$. Here the blue colors represent roots of the respective trees.
  • Figure 3.3: A sin-tree $\mathcal{T}_\infty$, namely a tree rooted at $0$ with a single infinite path to infinity, and the corresponding extended fringe $F_3(0,\mathcal{T}_\infty)$ upto level three about $0$.
  • Figure 3.4: The process $\mathop{\mathrm{BP}}\nolimits_{\alpha}(\cdot)$ in continuous time starting from the root $\rho$ and stopped at $\tau_{15}$.
  • ...and 9 more figures

Theorems & Definitions (23)

  • Definition 2.1: PageRank scores with damping factor $c$
  • Definition 3.1
  • Definition 3.2: Erdős-Rényi random graph model
  • Definition 3.3: $\mathbb{G}_{\star}$ as a metric space
  • Definition 3.4: Local weak convergence and the standard construction
  • Theorem 3.5: van2023random
  • Definition 3.6: Local weak convergence in the probability sense
  • Definition 3.7: Local weak convergence
  • Definition 3.8: Fringe distribution aldous-fringe
  • Lemma 3.9
  • ...and 13 more