On Range Matrices and Wireless Networks in d Dimensions

  • Authors:
  • Madhav Desai;D. Manjunath

  • Affiliations:
  • Indian Institute of Technology - Bombay;Indian Institute of Technology - Bombay

  • Venue:
  • WIOPT '05 Proceedings of the Third International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks
  • Year:
  • 2005

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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 驴 i,j 驴 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?"