Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Moderate deviations of triangle counts in sparse Erdős-Rényi random graphs $G(n,m)$ and $G(n,p)$

José D. Alvarado, Leonardo Gonçalves de Oliveira, Simon Griffiths

TL;DR

This work resolves moderate deviation probabilities for triangle counts in sparse Erdős–Rényi graphs G(n,m) and G(n,p), identifying the deviation scale that yields probability e^{-b} across a broad density range. The authors develop a martingale framework with increments tied to cherry and codegree structure, apply Freedman-type concentration, and carefully partition regimes into normal, star, hub, and clique localisation. They provide both upper and lower bounds that match up to constants in the non-localised regime and extend results to lower tails and cherry counts. The methods illuminate how local subgraph structures and global edge-density interact to drive deviations, with implications for understanding typical and atypical subgraph counts in sparse random graphs and potential generalisations to other subgraphs.

Abstract

We consider the question of determining the probability of triangle count deviations in the Erdős-Rényi random graphs $G(n,m)$ and $G(n,p)$ with densities larger than $n^{-1/2}(\log{n})^{1/2}$. In particular, we determine the log probability $\log\mathbb{P}(N_{\triangle}(G)\, >\, (1+δ)p^3n^3)$ up to a constant factor across essentially the entire range of possible deviations, in both the $G(n,m)$ and $G(n,p)$ model. For the $G(n,p)$ model we also prove a stronger result, up to a $(1+o(1))$ factor, in the non-localised regime. We also obtain some results for the lower tail and for counts of cherries (paths of length $2$).

Moderate deviations of triangle counts in sparse Erdős-Rényi random graphs $G(n,m)$ and $G(n,p)$

TL;DR

This work resolves moderate deviation probabilities for triangle counts in sparse Erdős–Rényi graphs G(n,m) and G(n,p), identifying the deviation scale that yields probability e^{-b} across a broad density range. The authors develop a martingale framework with increments tied to cherry and codegree structure, apply Freedman-type concentration, and carefully partition regimes into normal, star, hub, and clique localisation. They provide both upper and lower bounds that match up to constants in the non-localised regime and extend results to lower tails and cherry counts. The methods illuminate how local subgraph structures and global edge-density interact to drive deviations, with implications for understanding typical and atypical subgraph counts in sparse random graphs and potential generalisations to other subgraphs.

Abstract

We consider the question of determining the probability of triangle count deviations in the Erdős-Rényi random graphs and with densities larger than . In particular, we determine the log probability up to a constant factor across essentially the entire range of possible deviations, in both the and model. For the model we also prove a stronger result, up to a factor, in the non-localised regime. We also obtain some results for the lower tail and for counts of cherries (paths of length ).
Paper Structure (34 sections, 38 theorems, 318 equations, 3 figures)

This paper contains 34 sections, 38 theorems, 318 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Our results for triangle counts in $G(n,m)$
  3. Results for triangle counts in $G(n,p)$
  4. Our results for cherry counts
  5. An informal discussion of our approach
  6. Overview of the article
  7. Important auxiliary results and inequalities
  8. Chernoff inequalities and related inequalities
  9. Martingale deviation inequalities
  10. The martingale representations of the deviations of cherries and triangles
  11. A bound on a certain summation
  12. Degrees in $G(n,m)$
  13. The increments $X_{\wedge}(G_i)$
  14. Upper bound on cherry count deviation in $G(n,m)$
  15. Bounding the probability that $D'_{\wedge}(G_m)\, \geqslant \, C_1M_{\wedge}(b,t)$
  16. ...and 19 more sections

Key Result

Theorem 1.1

There exist absolute constants $c,C$ such that the following holds. For all $Cn^{-1/2}(\log{n})^{1/2}\leqslant t\leqslant 1/2$ and $3\log{n}\leqslant b\leqslant tn^2\ell$ we have

Figures (3)

  • Figure 1: In this figure, grey represents very small deviations, and the purple line ($\theta=0$) corresponds to the traditional large deviation results. Each of the other colours represents a different "most likely cause" of the corresponding deviation. In the green region: Good luck without a structural cause. In the light blue region: A star. In the dark blue region: A hub. In the red regions: A clique.
  • Figure 2: As in the previous figure, the purple line corresponds to the traditional large deviation results. Each of the other colours represents a different "most likely cause" of the corresponding deviation. In the green region: Good luck without a structural cause. In the light blue region: A star. In the dark blue region: A hub. In the red regions: A clique.
  • Figure 3: The result of Döring and Eichelsbacher DE applies in the light green region. We extend the same bound to the entire "normal" regime, shown here in green, see Theorem \ref{['thm:mainp']}. In the three coloured regions on the right, localisation (described below) occurs. The three colours correspond to the corresponding type of localisation, light blue for a star, dark blue for a hub and red for a clique. We remark that the grey region corresponds to deviations smaller than the order of the standard deviation, which have probability $\frac{1}{2}+o(1)$ by the central limit theorem. (The traditional large deviation results lie in the purple line along the right hand side.)

Theorems & Definitions (64)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Lemma 2.1
  • Lemma 2.2: Freedman's inequality
  • ...and 54 more