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Large deviation principle for the norm of the Laplacian matrix of inhomogeneous Erdős-Rényi random graphs

Rajat Subhra Hazra, Frank den Hollander, Maarten Markering

TL;DR

This work establishes large deviation principles for the largest eigenvalue of the Laplacian of inhomogeneous Erdős–Rényi graphs converging to a graphon limit. By leveraging the LDP for the empirical graphon and the graphon-operator framework, the authors derive a downward LDP for the normalized Laplacian norm $\lambda_N/N$ with rate ${N\choose 2}$ and a corresponding rate function $\psi_r$, as well as an upward LDP with rate $N$ and rate function $\widehat{\psi}_r$; the rate functions are characterized variationally via the basic graphon-relative-entropy functional $I_r$ and a single-vertex degree LDP with rate function $J_r(x,\beta)$. The analysis hinges on the nontrivial interplay between graphon convergence, Laplacian operator properties, and variational principles, including a careful contraction and change-of-measure technique to derive lower bounds and scaling near the minimizers. The results illuminate how spectral deviations of dense inhomogeneous random graphs are governed by degree fluctuations and provide a rigorous framework for understanding extreme Laplacian behavior in graphon-structured random graphs, with potential implications for network robustness and diffusion processes on random graphs.

Abstract

We consider an inhomogeneous Erdős-Rényi random graph $G_N$ with vertex set $[N] = \{1,\dots,N\}$ for which the pair of vertices $i,j \in [N]$, $i\neq j$, is connected by an edge with probability $r_N(\tfrac{i}{N},\tfrac{j}{N})$, independently of other pairs of vertices. Here, $r_N\colon\,[0,1]^2 \to (0,1)$ is a symmetric function that plays the role of a reference graphon. Let $λ_N$ be the maximal eigenvalue of the Laplacian matrix of $G_N$. We show that if $\lim_{N\to\infty} \|r_N-r\|_\infty = 0$ for some limiting graphon $r\colon\,[0,1]^2 \to (0,1)$, then $λ_N/N$ satisfies a downward LDP with rate $\binom{N}{2}$ and an upward LDP with rate $N$. We identify the associated rate functions $ψ_r$ and $\widehatψ_r$, and derive their basic properties.

Large deviation principle for the norm of the Laplacian matrix of inhomogeneous Erdős-Rényi random graphs

TL;DR

This work establishes large deviation principles for the largest eigenvalue of the Laplacian of inhomogeneous Erdős–Rényi graphs converging to a graphon limit. By leveraging the LDP for the empirical graphon and the graphon-operator framework, the authors derive a downward LDP for the normalized Laplacian norm with rate and a corresponding rate function , as well as an upward LDP with rate and rate function ; the rate functions are characterized variationally via the basic graphon-relative-entropy functional and a single-vertex degree LDP with rate function . The analysis hinges on the nontrivial interplay between graphon convergence, Laplacian operator properties, and variational principles, including a careful contraction and change-of-measure technique to derive lower bounds and scaling near the minimizers. The results illuminate how spectral deviations of dense inhomogeneous random graphs are governed by degree fluctuations and provide a rigorous framework for understanding extreme Laplacian behavior in graphon-structured random graphs, with potential implications for network robustness and diffusion processes on random graphs.

Abstract

We consider an inhomogeneous Erdős-Rényi random graph with vertex set for which the pair of vertices , , is connected by an edge with probability , independently of other pairs of vertices. Here, is a symmetric function that plays the role of a reference graphon. Let be the maximal eigenvalue of the Laplacian matrix of . We show that if for some limiting graphon , then satisfies a downward LDP with rate and an upward LDP with rate . We identify the associated rate functions and , and derive their basic properties.
Paper Structure (32 sections, 12 theorems, 115 equations, 2 figures)

This paper contains 32 sections, 12 theorems, 115 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. Background
  3. LDP for inhomogeneous Erdős-Rényi random graphs
  4. Graphons
  5. LDP for empirical graphon
  6. Graphon operators
  7. Main theorems: downward large deviations
  8. Main theorems: upward large deviations
  9. Discussion
  10. Proof of Proposition \ref{['prop:basics']}
  11. Lower semi-continuity
  12. Proof of Proposition \ref{['prop:basics']}
  13. Proof of theorems for downward LDP
  14. Proof of Theorem \ref{['thm:main']}
  15. Upper bound
  16. ...and 17 more sections

Key Result

Theorem 1.1

[LDP for inhomogeneous ERRG, DS19, M20] Subject to Assbasic--Assbasic2, the sequence $(\widetilde{\mathbb{P}}_N)_{N\in\mathbb{N}}$ satisfies the large deviation principle on $(\widetilde{\mathcal{W}},\delta_{\square})$ with rate $\binom{N}{2}$, i.e., where the rate function $J_r\colon\,\widetilde{\mathcal{W}} \to \mathbb{R}$ is given by where $h$ is any representative of $\widetilde{h}$ and wit

Figures (2)

  • Figure 1: Graph of $\beta \mapsto \psi_r(\beta)$.
  • Figure 2: Graph of $\beta \mapsto \widehat{\psi}_r(\beta)$.

Theorems & Definitions (23)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 13 more