## Connectivity analysis of wireless ad-hoc networks

##### Abstract

Connectivity is one of the most fundamental properties of wireless
ad-hoc networks as most network functions are predicated upon the
network being connected. Although increasing node transmission power
will improve network connectivity, too large a power level is not
feasible as energy is a scarce resource in wireless ad-hoc networks.
Thus, it is crucial to identify the minimum node transmission power
that will ensure network connectivity with high probability.
It is known that there exists a critical level transmission power
such that a suitably larger power will ensure network connectivity
with high probability. A small variation across this threshold level
will lead to a sharp transition of the probability that the network
is connected. Thus, in order to precisely estimate the minimum node
transmission power, not only do we need to identify this critical
threshold, but also how fast this transition takes place. To
characterize the sharpness of transition, we define weak, strong and
very strong critical thresholds associated with increasing
transition speeds.
In this dissertation, we seek to estimate the minimum node
transmission power for large scale one-dimensional wireless ad-hoc
networks under the Geometric Random Graph (GRG) models. Unlike in
previous works where nodes are taken to be uniformly distributed, we
assume a more general node distribution. Using the methods of first
and second moments, we theoretically prove the existence of a very
strong critical threshold when the density function is everywhere
positive. On the other hand, only weak thresholds are shown to exist
when the density function contains vanishing densities.
We also study the connectivity of two-dimensional wireless ad-hoc
networks under the random connection model, which accounts for
statistical channel variations. With the help of the Stein-Chen
method, we derive a closed form formula for the limiting probability
that there are no isolated nodes under a very general assumption of
channel variations. The node transmission power to ensure the
absence of isolated nodes provides a tight lower bound on the
transmission power needed to ensure network connectivity.

University of Maryland, College Park, MD 20742-7011 (301)314-1328.

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