Let n be a positive integer, and λ > 0 a real number. Let Vn be a set of n points randomly located within the unit disk, which are mutually independent. For 1 ≤ p ≤ ∞, define Gp(λ, n) to be the graph with the vertex set Vn, in which two vertices are adjacent if and only if their `p-distance is at most λ. We call this graph a unit disk random graph. Let λ = c √ ln n/n and let X be the number of ...

We investigate the problem of disseminating broadcast messages in wireless networks with time-varying links from a percolation-based perspective. Using a model of wireless networks based on random geometric graphs with dynamic on-off links, we show that the delay for disseminating broadcast information exhibits two behavioral regimes, corresponding to the phase transition of the underlying netw...

The random walk with choice is a well known variation to the random walk that first selects a subset of d neighbours nodes and then decides to move to the node which maximizes the value of a certain metric; this metric captures the number of (past) visits of the walk to the node. In this paper we propose an enhancement to the random walk with choice by considering a new metric that captures not...

This paper undergoes quantitative analyses on fundamental properties of ad hoc networks including estimating the number of hiddenterminal pairs and the number of exposed-terminal sets. To obtain these results, we propose a paradigm to systematically derive exact formulas for a great deal of subgraph probabilities of random geometric graphs. In contrast to previous work, which established asympt...

The diversity of multimedia devices is an important feature on the Internet, which demands an executable mechanism to intelligently adjust the appearance of a document according to different viewing contexts. In order to satisfy this requirement, a graph-grammar-based approach for multimedia authoring is presented by extending a context-sensitive graph grammar formalism, Reserved Graph Grammar ...

We theoretically and experimentally analyze the process of adding sparse random links to a random wireless networks modeled as a random geometric graph. While this process has been previously proposed, we are the first to (i) theoretically consider sparse new-wiring (with asymptotically less than constant fraction of nodes allowed very small constant new wired edges), (ii) prove bounds for any ...

LetP be a Poisson process of intensity one in a squareSn of arean. We construct a random geometric graph Gn,k by joining each point of P to its k ≡ k(n) nearest neighbours. Recently, Xue and Kumar proved that if k ≤ 0.074 log n then the probability that Gn,k is connected tends to 0 as n → ∞ while, if k ≥ 5.1774 log n, then the probability that Gn,k is connected tends to 1 as n → ∞. They conject...

A random geometric graph Gn is constructed by taking vertices X1, . . . , Xn ∈ R at random (i.i.d. according to some probability distribution ν with a bounded density function) and including an edge between Xi and Xj if ‖Xi−Xj‖ < r where r = r(n) > 0. We prove a conjecture of Penrose ([14]) stating that when r = r(n) is chosen such that nr = o(lnn) then the probability distribution of the cliqu...

We consider three classes of random graphs: edge random graphs, vertex random graphs, and vertex-edge random graphs. Edge random graphs are Erd˝ os-Rényi random graphs [5, 6], vertex random graphs are generalizations of geometric random graphs [16], and vertex-edge random graphs generalize both. The names of these three types of random graphs describe where the randomness in the models lies: in...