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Large deviation principles for graphon sampling

Jan Grebík, Oleg Pikhurko

TL;DR

This work analyzes large deviation principles for sampling from a graphon W via G(n,W) across different exponential speeds, identifying precisely which speeds yield an LDP and giving explicit rate functions. It introduces and studies central rate-function constructs K_W, J_{α,p}, and R_p, and leverages reductions to weighted graph sampling and stochastic block models to obtain LDPs for step-graphon and colored-graphon settings. The results unify and extend the Chatterjee–Varadhan framework to inhomogeneous graphons, quantify deviations in weighted and block-structured models, and establish exponential equivalence to transfer LDPs between models. The paper also develops foundational continuity properties and closes with outlook on the remaining challenge of LDPs at speed Θ(n^2) for general graphons, highlighting structural conditions like RN(W) and FORB(W).

Abstract

We investigate possible large deviation principles (LDPs) for the $n$-vertex sampling from a given graphon with various speeds $s(n)$ and resolve all the cases except when the speed $s(n)$ is of order $n^2$. For quadratic speed $s=(c+o(1))n^2$, we establish an LDP for an arbitrary $k$-step graphon, which extends a result of Chatterjee and Varadhan [Europ. J. Combin., 32 (2011) 1000-1017] who did this for $k=1$ (that is, for the homogeneous binomial random graphs). This is done by reducing the problem to the LDP for stochastic $k$-block models established recently by Borgs, Chayes, Gaudio, Petti and Sen ["A large deviation principle for block models", arxiv:2007.14508, 2020]. Also, we improve some results by Borgs et al.

Large deviation principles for graphon sampling

TL;DR

This work analyzes large deviation principles for sampling from a graphon W via G(n,W) across different exponential speeds, identifying precisely which speeds yield an LDP and giving explicit rate functions. It introduces and studies central rate-function constructs K_W, J_{α,p}, and R_p, and leverages reductions to weighted graph sampling and stochastic block models to obtain LDPs for step-graphon and colored-graphon settings. The results unify and extend the Chatterjee–Varadhan framework to inhomogeneous graphons, quantify deviations in weighted and block-structured models, and establish exponential equivalence to transfer LDPs between models. The paper also develops foundational continuity properties and closes with outlook on the remaining challenge of LDPs at speed Θ(n^2) for general graphons, highlighting structural conditions like RN(W) and FORB(W).

Abstract

We investigate possible large deviation principles (LDPs) for the -vertex sampling from a given graphon with various speeds and resolve all the cases except when the speed is of order . For quadratic speed , we establish an LDP for an arbitrary -step graphon, which extends a result of Chatterjee and Varadhan [Europ. J. Combin., 32 (2011) 1000-1017] who did this for (that is, for the homogeneous binomial random graphs). This is done by reducing the problem to the LDP for stochastic -block models established recently by Borgs, Chayes, Gaudio, Petti and Sen ["A large deviation principle for block models", arxiv:2007.14508, 2020]. Also, we improve some results by Borgs et al.
Paper Structure (21 sections, 27 theorems, 129 equations)

This paper contains 21 sections, 27 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Overview of the new results
  3. Large deviations for general topological spaces
  4. Binomial random graph and graphons
  5. Sampling from a graphon
  6. LDP for step graphons for speed $\Theta(n^2)$
  7. Stochastic block models
  8. Organisation of the paper
  9. Preliminaries
  10. Graphons
  11. Topology
  12. Properties of the function $K_W$
  13. LDPs for $(\widetilde{\mathbb{H}}_{n,W})_{n\in {\mathbbm N}}$
  14. Exponential equivalence
  15. LDPs for the measures $\widetilde{\mathbb{R}}_{n,W}$
  16. ...and 6 more sections

Key Result

Theorem 1.2

Let $p\in [0,1]$. The function $I_p:{\mathcal{W}}\to [0,\infty]$ defined by gives a well-defined function $\widetilde{\mathcal{W}}\to [0,\infty]$ (that is, $I_p$ assumes the same value at any two graphons at $\delta_\Box$-distance 0) which is lower semi-continuous on $(\widetilde{\mathcal{W}},\delta_\Box)$. Moreover, the sequence of measures $(\widetilde{\mathbb{P}}_{n,p})_

Theorems & Definitions (57)

  • Definition 1.1
  • Theorem 1.2: Chatterjee and Varadhan ChatterjeeVaradhan11
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5: Borgs, Chayes, Gaudio, Petti and Sen BCGPS
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • ...and 47 more