Hostname: page-component-77c89778f8-swr86 Total loading time: 0 Render date: 2024-07-18T16:58:24.234Z Has data issue: false hasContentIssue false

A Large Deviation Result on the Number of Small Subgraphs of a Random Graph

Published online by Cambridge University Press:  12 April 2001

Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA (e-mail:


Fix a small graph H and let YH denote the number of copies of H in the random graph G(n, p). We investigate the degree of concentration of YH around its mean, motivated by the following questions.

[bull ] What is the upper tail probability Pr(YH [ges ] (1 + ε)[ ](YH))?

[bull ] For which λ does YH have sub-Gaussian behaviour, namely

(formula here)

where c is a positive constant?

[bull ] Fixing λ = ω(1) in advance, find a reasonably small tail T = T(λ) such that

(formula here)

We prove a general concentration result which contains a partial answer to each of these questions. The heart of the proof is a new martingale inequality, due to J. H. Kim and the present author [13].

Research Article
2001 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)