A random geometric irrigation graph Γn(rn, ξ) has n vertices identified by n independent uniformly distributed points X1, . . . , Xn in the unit square [0, 1]. Each point Xi selects ξi neighbors at random, without replacement, among those points Xj (j 6= i) for which ‖Xi − Xj‖ < rn, and the selected vertices are connected to Xi by an edge. The number ξi of the neighbors is an integer-valued ran...

We show new lower bounds for collision-free transmissions in Radio Networks. Our main result is a tight lower bound of Ω(log n log(1/ )) on the time required by a uniform randomized protocol to achieve a clear transmission with success probability 1 − in a one-hop setting. This result is extended to non-uniform protocols as well. A new lower bound is proved for the important multi-hop setting o...

The Reserved Graph Grammar (RGG) is a general graph grammar Jormalism that expresses a wide range oj visual languages, This paper presents an extension to RGG with the capability oj spatial specification, Graph transJormation satisfying the spatial specification can be performed in the process oj parsing. The RGG with spatial specification can be applied to various types oj applications, The pa...

We are interested in the problem of computing the average consensus in a distributed fashion on random geometric graphs. We describe a new algorithm called Multi-scale Gossip which employs a hierarchical decomposition of the graph to partition the computation into tractable sub-problems. Using only pairwise messages of fixed size that travel at most O(n 1 3 ) hops, our algorithm is robust and h...

We give a general local central limit theorem for the sum of two independent random variables, one of which satisfies a central limit theorem while the other satisfies a local central limit theorem with the same order variance. We apply this result to various quantities arising in stochastic geometry, including: size of the largest component for percolation on a box; number of components, numbe...

We examine maximum vertex coloring of random geometric graphs, in an arbitrary but fixed dimension, with a constant number of colors. Since this problem is neither scaleinvariant nor smooth, the usual methodology to obtain limit laws cannot be applied. We therefore leverage different concepts based on subadditivity to establish convergence laws for the maximum number of vertices that can be col...

We study two geometric models of random k-satisfiability which combine random k-SAT with the Random Geometric Graph: boolean literals are placed uniformly at random or according to a Poisson process in a cube, and for each set of k literals contained in a ball of a given radius, a clause is formed. For k = 2 we find the exact location of the satisfiability threshold (as either the radius or int...