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On the upper tail of star counts in random graphs

Margarita Akhmejanova, Matas Šileikis

TL;DR

This work determines the exponential decay rate of the upper tail for the number of $r$-stars in $\,\mathbb{G}(n,p)$ in the sparse regime $p\to 0$ with $\mu=\mathbb{E}X\to\infty$, revealing a transition among Poisson, localization, and intermediate regimes. The authors reduce the problem to an i.i.d. binomial model via McKay–Wormald graph enumeration and analyze the tail of $Y=\sum_{i=1}^n {X_i \choose r}$ by splitting into small and large contributions, governed by a variational rate $I_r(c,\varepsilon)$ and its minimizers. Their main result provides precise asymptotics for $-\log \mathbb{P}(X \ge (1+\varepsilon)\mu)$ in regimes where $\mu/\log^{r/(r-1)} n \to c$ and where $\mu$ grows faster, thereby solving a sparse instance of the upper-tail problem for an irregular graph $H=K_{1,r}$. This advances the understanding of large deviations in random graphs, extends prior sparse-regime results for regular graphs, and offers a framework for analyzing convex-statistics of degree sequences in random graphs.

Abstract

Let $X$ count the number of $r$-stars in the random binomial graph $\mathbb{G}(n,p)$. We determine, for fixed $r$ and $\varepsilon > 0$, the asymptotics of $\log \mathbb{P}(X \ge (1 + \varepsilon)\mathbb{E} X)$ assuming only $\mathbb{E} X \to \infty$ and $p \to 0$ thus giving a first class of irregular graphs for which the upper tail problem for subgraph counts (stated by Janson and Ruciński in 2004) is solved in the sparse setting.

On the upper tail of star counts in random graphs

TL;DR

This work determines the exponential decay rate of the upper tail for the number of -stars in in the sparse regime with , revealing a transition among Poisson, localization, and intermediate regimes. The authors reduce the problem to an i.i.d. binomial model via McKay–Wormald graph enumeration and analyze the tail of by splitting into small and large contributions, governed by a variational rate and its minimizers. Their main result provides precise asymptotics for in regimes where and where grows faster, thereby solving a sparse instance of the upper-tail problem for an irregular graph . This advances the understanding of large deviations in random graphs, extends prior sparse-regime results for regular graphs, and offers a framework for analyzing convex-statistics of degree sequences in random graphs.

Abstract

Let count the number of -stars in the random binomial graph . We determine, for fixed and , the asymptotics of assuming only and thus giving a first class of irregular graphs for which the upper tail problem for subgraph counts (stated by Janson and Ruciński in 2004) is solved in the sparse setting.
Paper Structure (13 sections, 12 theorems, 122 equations, 1 figure)

This paper contains 13 sections, 12 theorems, 122 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Localisation: comparison with regular graph counts
  3. Plan of the proof
  4. Preliminaries
  5. Tail bounds
  6. Asymptotic enumeration of graphs with a given sequence
  7. A basic property of convex functions
  8. Reduction to i.i.d. binomials
  9. Upper tail for i.i.d. binomials
  10. Small values
  11. Large values
  12. The whole sum in the sparse case
  13. Proof of Theorem \ref{['thm:stars_UT']}

Key Result

Theorem 1

Fix an integer $r \ge 2$ and let $X = X^{K_{1,r}}_{n,p}$ be the number of copies of star $K_{1,r}$ in the random graph $\mathbb{G}(n,p)$. For $\delta \ge 0$ define a function $\psi_r(\delta) = r^{-1}(r!\delta)^{1/r}$ and let Furthermore, set $I_r(0, \varepsilon) = \phi(\varepsilon)$. If $p = p(n) \to 0$ is such that $\mu \gg \log n$, then

Figures (1)

  • Figure 1: Case $r = 2, \varepsilon = 1$. Left: graphs of $f_a$ for a few values of $a$; right: graphs of $\log(1 + \varepsilon - \delta)$ as well as $(a/r)\delta^{-1/r}$.

Theorems & Definitions (23)

  • Theorem 1
  • Remark 1
  • Lemma 2
  • proof
  • Theorem 3: SW20*Theorem 7
  • Proposition 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 13 more