József Balogh

On k-coverage in a mostly sleeping sensor network

Sensor networks are often desired to last many times longer than the active lifetime of individual sensors. This is usually achieved by putting sensors to sleep for most of their lifetime. On the other hand, surveillance kind of applications require guaranteed k-coverage of the protected region at all times. As a result, determining the appropriate number of sensors to deploy that achieves both goals simultaneously becomes a challenging problem. In this paper, we consider three kinds of deployments for a sensor network on a unit square - a √n x √n grid, random uniform (for all n points), and Poisson (with density n). In all three deployments, each sensor is active with probability p, independently from the others. Then, we claim that the critical value of the function npπr2/log(np) is 1 for the event of k-coverage of every point. We also provide an upper bound on the window of this phase transition. Although the conditions for the three deployments are similar, we obtain sharper bounds for the random deployments than the grid deployment, which occurs due to the boundary condition. In this paper, we also provide corrections to previously published results for the grid deployment model. Finally, we use simulation to show the usefulness of our analysis in real deployment scenarios.

Bootstrap percolation in three dimensions.

By bootstrap percolation we mean the following deterministic process on a graph G. Given a set A of vertices infected at time 0, new vertices are subsequently infected, at each time step, if they have at least ∈ N previously infected neighbors. When the set A is chosen at random, the main aim is to determine the critical probability p c (G, r) at which percolation (infection of the entire graph) becomes likely to occur. This bootstrap process has been extensively studied on the d-dimensional grid [n] d : with 2 ≤ r ≤ d fixed, it was proved by Cerf and Cirillo (for d = r = 3), and by Cerf and Manzo (in general), that p c ([n] d ,r)=Θ(1/log (r-1) n) d-r+1 , where log (r) is an r-times iterated logarithm. However, the exact threshold function is only known in the case d = r = 2, where it was shown by Holroyd to be (1 + o(1)) π 2 18 log n. In this paper we shall determine the exact threshold in the crucial case d = r = 3, and lay the groundwork for solving the problem for all fixed d and r.

Bootstrap Percolation on Infinite Trees and Non-Amenable Groups

Bootstrap percolation on an arbitrary graph has a random initial configuration, where each vertex is occupied with probability

Covering planar graphs with forests

p

A jump to the bell number for hereditary graph properties

, independently of each other, and a deterministic spreading rule with a fixed parameter

Hamilton cycles in random geometric graphs

k

Sharp thresholds in Bootstrap percolation

: if a vacant site has at least

A remark on the number of edge colorings of graphs

k

Index assignment for two-channel quantization

occupied neighbours at a certain time step, then it becomes occupied in the next step. This process is well studied on

Private computation using a PEZ dispenser

{mathbb Z}^d