Consider a 2-dimensional soft random geometric graph G(λ,s,ϕ), obtained by placing Poisson(λs2) number of vertices uniformly at in square side s, with edges placed between each pair x,y probability ϕ(‖x−y‖), where ϕ: R+→[0,1] is finite-range connection function. This paper concerned the asymptotic behaviour G(λ,s,ϕ) large-s limit (λ,ϕ) fixed. We prove that proportion largest component converges...

The efficient parsing algorithms are usually confined to confluent graph grammars in context-sensitive formalisms, whereas the confluence condition is not frequently met by graph grammars in practical applications. This paper, based on the Reserved Graph Grammar (RGG) formalism, proposes an extended formalism XRGG which allows imposing constraints characterizing application situations on graph ...

Wireless networks in which the node locations are random are best modelled as random geometric graphs (RGGs). In addition to their extensive application in the modelling of wireless networks, RGGs are finding many new applications and are being studied in their own right. In this paper we first provide a brief introduction to the issues of interest in random wireless networks. We then discuss s...

To capture the inherent geometric features of many community detection problems, we propose to use a new random graph model of communities that we call a Geometric Block Model. The geometric block model generalizes the random geometric graphs in the same way that the well-studied stochastic block model generalizes the Erdös-Renyi random graphs. It is also a natural extension of random community...

We propose a coloring algorithm for sparse random graphs generated by the geographical threshold graph (GTG) model, a generalization of random geometric graphs (RGG). In a GTG, nodes are distributed in a Euclidean space, and edges are assigned according to a threshold function involving the distance between nodes as well as randomly chosen node weights. The motivation for analyzing this model i...

We study the behavior of random geometric graphs in high dimensions. We show that as the dimension grows, the graph becomes similar to an Erdős-Rényi random graph. We pay particular attention to the clique number of such graphs and show that it is very close to that of the corresponding Erdős-Rényi graph when the dimension is larger than log n where n is the number of vertices. The problem is m...

We study the normal approximation of functionals of Poisson measures having the form of a finite sum of multiple integrals. When the integrands are nonnegative, our results yield necessary and sufficient conditions for central limit theorems. These conditions can always be expressed in terms of contraction operators or, equivalently, fourth cumulants. Our findings are specifically tailored to d...