Sensitivity of critical transmission ranges to node placement distributions

Authors:
Guang Han;Armand M. Makowski
Affiliations:
Networks Advanced Technologies, Motorola Inc., Arlington Heights, IL and Department of Electrical and Computer Engineering and the Institute for Systems Research, University of Maryland, College P ...;Department of Electrical and Computer Engineering, and the Institute for Systems Research, University of Maryland, College Park, MD
Venue:
IEEE Journal on Selected Areas in Communications - Special issue on stochastic geometry and random graphs for the analysis and designof wireless networks
Year:
2009

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Abstract

We consider the geometric random graph where n points are distributed independently on the unit interval [0, 1] according to some probability distribution function F with density f. Two nodes are adjacent (i.e., communicate with each other) if their distance is less than some transmission range. We survey results, some classical and some recently obtained by the authors, concerning the existence of zero-one laws for graph connectivity, the type of zero-one laws under the specific assumptions made, the form of its critical scaling and its dependence on the density f. We also present results and conjectures concerning the width of the corresponding phase transition. Engineering implications are discussed for power allocation.