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The giant in random graphs is almost local

Remco van der Hofstad

TL;DR

The paper addresses how local convergence of sparse random graphs governs the emergence and size of the giant component. It introduces a simple 'giant is almost local' criterion that, together with local convergence in probability, guarantees convergence of the giant's size to the local limit's survival probability $\zeta$ and describes the giant's local structure via the limit. The approach relies on coupling graph exploration with $n$-dependent unimodular branching processes, yielding LLNs for the giant in the configuration model and in spatially inhomogeneous graphs (GIRGs), and also reproving small-world distances. These results bridge local limit theory with global giant behavior, providing tools for studying percolation and other global properties from local data.

Abstract

Local convergence techniques have become a key methodology to study sparse random graphs. However, convergence of many random graph properties does not directly follow from local convergence. A notable, and important, such random graph property is the size and uniqueness of the giant component. We provide a simple criterion that guarantees that local convergence of a random graph implies the convergence of the proportion of vertices in the maximal connected component. We further show that, when this condition holds, the local properties of the giant, as well as its complement, are also described by the local limit. We give several examples where this method gives rise to a novel law of large numbers for the giant, based on results proved in the literature. Aside from these examples, we apply our method to the classical problem of giants in the configuration model as a proof of concept, reproving a well-established result. As a side result of this proof, we give an extremely simple proof of the small-world nature of the configuration model.

The giant in random graphs is almost local

TL;DR

The paper addresses how local convergence of sparse random graphs governs the emergence and size of the giant component. It introduces a simple 'giant is almost local' criterion that, together with local convergence in probability, guarantees convergence of the giant's size to the local limit's survival probability and describes the giant's local structure via the limit. The approach relies on coupling graph exploration with -dependent unimodular branching processes, yielding LLNs for the giant in the configuration model and in spatially inhomogeneous graphs (GIRGs), and also reproving small-world distances. These results bridge local limit theory with global giant behavior, providing tools for studying percolation and other global properties from local data.

Abstract

Local convergence techniques have become a key methodology to study sparse random graphs. However, convergence of many random graph properties does not directly follow from local convergence. A notable, and important, such random graph property is the size and uniqueness of the giant component. We provide a simple criterion that guarantees that local convergence of a random graph implies the convergence of the proportion of vertices in the maximal connected component. We further show that, when this condition holds, the local properties of the giant, as well as its complement, are also described by the local limit. We give several examples where this method gives rise to a novel law of large numbers for the giant, based on results proved in the literature. Aside from these examples, we apply our method to the classical problem of giants in the configuration model as a proof of concept, reproving a well-established result. As a side result of this proof, we give an extremely simple proof of the small-world nature of the configuration model.

Paper Structure

This paper contains 44 sections, 14 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Asymptotics and properties of the giant
  3. Notation.
  4. Local convergence in probability
  5. Asymptotics of the giant
  6. Local properties of the giant
  7. Examples through cluster tail bounds outside giant
  8. Application to spatial inhomogeneous random graphs.
  9. The 'giant is almost local' condition revisited
  10. Application to the configuration model
  11. Model definition and results
  12. Model definition and assumptions.
  13. The giant in the configuration model.
  14. Local structure of $\mathscr{C}_{\rm max}$.
  15. Local convergence of configuration models.
  16. ...and 29 more sections

Key Result

Corollary 2.1

Let $(G_n)_{n\geq 1}$ be a sequence of graphs having size $|V(G_n)|=n$. Assume that $G_n$ converges locally in probability to $(G,o)\sim \mu$. Write $\zeta=\mu(|\mathscr{C}(o)|=\infty)$ for the survival probability of the limiting graph $(G,o)$. Then, for every $\varepsilon>0$ fixed, as $n\rightarro

Theorems & Definitions (20)

  • Corollary 2.1: Upper bound on the giant
  • Theorem 2.2: The giant is almost local
  • Remark 2.3: Proof of necessity of \ref{['almost-local-cond']}
  • Theorem 2.4: Properties of the giant
  • Theorem 2.5: Local limit of the giant
  • Corollary 2.6: Law of large numbers giant given bound components outside it
  • Theorem 2.7: Giant in GIRGs
  • Remark 2.8: Relation to second largest component
  • Lemma 2.9: Condition \ref{['almost-local-cond']} revisited
  • Remark 2.10: Alternative 'giant is almost local' condition
  • ...and 10 more