A Tractable Framework for Exact Probability of Node Isolation and   Minimum Node Degree Distribution in Finite Multi-hop Networks
  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. 
Zubair Khalid, Salman Durrani and Jing Guo
cs.IT cs.NI math.IT