On the k-Connectivity of Ad-Hoc Wireless Networks

In this paper, we consider k-connectivity problem which is one of the fundamental problems in the ad-hoc wireless network. Here we assume that the nodes are uniformly distributed on a unit square area [0, 1]
²
. We define a random graph on unit square [0, 1]
²
and derive the critical range for transmission radius r
_n
to achieving k-connectivity of a randomly generated network in almost sure sense. We prove that if r
_n
is defined as r
_n
²
:=
¹
/
_nr
(c log n + (k-1)log
₂
n - logΓ(k)), where k, c>0 are constants and n be the number of node in the network, then for c>2 the network is a k-connected network in almost sure sense. Also we derive that for c>2 and critical transmission range C
_n
= √(
^{clog n}
/
_nr
) the network is connected.