In this paper, we analyze various critical transmitting/sensing ranges for connectivity and coverage in three-dimensional sensor networks. As in other large-scale complex systems, many global parameters of sensor networks undergo phase transitions: For a given property of the network, there is a critical threshold, corresponding to the minimum amount of the communication effort or power expenditure by individual nodes, above (resp. below) which the property exists with high (resp. a low) probability.For sensor networks, properties of interest include simple and multiple degrees of connectivity/coverage. First, we investigate the network topology according to the region of deployment, the number of deployed sensors and their transmitting/sensing ranges. More specifically, we consider the following problems:Assume that n nodes, each capable of sensing events within a radius of r, are randomly and uniformly distributed in a 3-dimensional region R of volume V , how large must the sensing range r Sense be to ensure a given degree of coverage of the region to monitor? For a given transmission range r Trans , what is the minimum (resp. maximum) degree of the network? What is then the typical hop-diameter of the underlying network? Next, we show how these results affect algorithmic aspects of the network by designing specific distributed protocols for sensor networks.

Let G(n, M ) be the uniform random graph with n vertices and M edges. Erdős and Rényi (1960) conjectured that the limiting probability) is planar} exists and is a constant strictly between 0 and 1. Luczak, Pittel and Wierman (1994) proved this conjecture and Janson, Luczak, Knuth and Pittel (1993) gave lower and upper bounds for this probability. In this paper we determine the exact limiting probability of a random graph being planar near the critical point M = n/2. For each λ, we find an exact analytic expression for(1 + λn −1/3 ) is planar .In particular, we obtain p(0) ≈ 0.99780. We extend these results to classes of graphs closed under taking minors. As an example, we show that the probability of G(n, n 2 ) being seriesparallel converges to 0.98003. For the sake of completeness and exposition we reprove in a concise way several basic properties we need of a random graph near the critical point.We dedicate this paper to the memory of Philippe Flajolet.

Abstract-In this work we consider a large-scale geographic area populated by tiny sensors and some more powerful devices called actors, authorized to organize the sensors in their vicinity into short-lived, actor-centric sensor networks. The tiny sensors run on miniature non-rechargeable batteries, are anonymous and are unaware of their location. The sensors differ in their ability to dynamically alter their sleep times. Indeed, the periodic sensors have sleep periods of predefined lengths, established at fabrication time; by contrast, the free sensors can dynamically alter their sleep periods, under program control. The main contribution of this work is to propose an energy-efficient location training protocol for heterogeneous actor-centric sensor networks where the sensors acquire coarse-grain location awareness with respect to the actor in their vicinity. Our analytical analysis, confirmed by experimental evaluation, show that the proposed protocol outperforms the best previously-known location training protocols in terms of the number of sleep/awake transitions, overall sensor awake time, and energy consumption.