## Advances in Applied Probability

- Adv. in Appl. Probab.
- Volume 37, Number 1 (2005), 1-24.

### Connectivity of random *k*-nearest-neighbour graphs

Paul Balister, Béla Bollobás, Amites Sarkar, and Mark Walters

#### Abstract

Let *P* be a Poisson process of intensity one in a square
*S*_{n} of area *n*. We construct a
random geometric graph *G*_{n,k} by
joining each point of *P* to its
*k* ≡ *k*(*n*) nearest neighbours. Recently,
Xue and Kumar proved that if *k* ≤ 0.074log*n*
then the probability that *G*_{n,k} is
connected tends to 0 as *n* → ∞ while, if
*k* ≥ 5.1774log*n*, then the probability that
*G*_{n,k} is connected tends to 1 as
*n* → ∞. They conjectured that the threshold
for connectivity is *k* = (1 + *o*(1))log*n*. In
this paper we improve these lower and upper bounds to
0.3043log*n* and 0.5139log*n*, respectively, disproving
this conjecture. We also establish lower and upper bounds of
0.7209log*n* and 0.9967log*n* for the directed version
of this problem. A related question concerns coverage. With
*G*_{n,k} as above, we surround each
vertex by the smallest (closed) disc containing its *k*
nearest neighbours. We prove that if
*k* ≤ 0.7209log*n* then the probability that
these discs cover *S*_{n} tends to 0 as
*n* → ∞ while, if
*k* ≥ 0.9967log*n*, then the probability that the
discs cover *S*_{n} tends to 1 as
*n* → ∞.

#### Article information

**Source**

Adv. in Appl. Probab. Volume 37, Number 1 (2005), 1-24.

**Dates**

First available in Project Euclid: 13 April 2005

**Permanent link to this document**

http://projecteuclid.org/euclid.aap/1113402397

**Digital Object Identifier**

doi:10.1239/aap/1113402397

**Mathematical Reviews number (MathSciNet)**

MR2135151

**Zentralblatt MATH identifier**

1079.05086

**Subjects**

Primary: 05C80: Random graphs [See also 60B20]

Secondary: 82B43: Percolation [See also 60K35]

**Keywords**

Random geometric graph connectivity Poisson process

#### Citation

Balister, Paul; Bollobás, Béla; Sarkar, Amites; Walters, Mark. Connectivity of random k -nearest-neighbour graphs. Adv. in Appl. Probab. 37 (2005), no. 1, 1--24. doi:10.1239/aap/1113402397. http://projecteuclid.org/euclid.aap/1113402397.