  This paper presents a tractable analytical framework for the exact
calculation of probability of node isolation and minimum node degree
distribution when $N$ sensor nodes are independently and uniformly distributed
inside a finite square region. The proposed framework can accurately account
for the boundary effects by partitioning the square into subregions, based on
the transmission range and the node location. We show that for each subregion,
the probability that a random node falls inside a disk centered at an arbitrary
node located in that subregion can be expressed analytically in closed-form.
Using the results for the different subregions, we obtain the exact probability
of node isolation and minimum node degree distribution that serves as an upper
bound for the probability of $k$-connectivity. Our theoretical framework is
validated by comparison with the simulation results and shows that the minimum
node degree distribution serves as a tight upper bound for the probability of
$k$-connectivity. The proposed framework provides a very useful tool to
accurately account for the boundary effects in the design of finite wireless
networks.
