Abstract
Suppose that V = {v_{1,} v_{2,} \ldots v_n} is a set of nodes randomly (uniformly) distributed in the d dimensional cube [0,x_0 ]^d, and W = {w(i,j) > 0 : 1 ≤ i,j ≤ n} is a set of numbers chosen so that w(i,j) = w(j,i). Construct a graph G_{n,d,w} whose vertex set is V, and whose edge set consists of all pairs {u_i ,u_j} with \left\| {\{ u_i - u_j \} } \right\| \leqslant w(i,j). In the wireless network context, the set V is a set of labeled nodes in the network and W represents the maximum distances between the node pairs for them to be connected. We essentially addressed the following question: "If G is a graph with vertex set V, what is the probability that G appears as a subgraph in G_{n,d,w}?" Our main contribution is a closed form expression for this probability under the \iota _\infty norm for any dimension d and a suitably defined probability density function. As a corollary to the above answer, we also answer the question, "What is the probability that G_{n,d,w} is connected?"