On range matrices and wireless networks in d dimensions

Abstract

Suppose that V = {v/sub 1/, v/sub 2/, ...v/sub n/} is a set of nodes randomly (uniformly) distributed in the d dimensional cube [0, x/sub 0/]/sup d/, and W = {w(i, j) > 0 : 1 /spl les/ i, j /spl les/ n} is a set of numbers chosen so that w(i, j) = w(j, i) = w(j, i). Construct a graph G/sub n,d,W/ whose vertex set is V, and whose edge set consists of all pairs {u/sub i/, u/sub j/} with /spl par/ u/sub i/ - u/sub j/ /spl par/ /spl les/ 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 address the following question: "if G is a graph with vertex set V, what is the probability that G appears as a subgraph in G/sub n,d,W/?" Our main contribution is a closed form expression for this probability under the l/sub /spl infin// 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 Q/sub n,d,W/ is connected?".