Causal Set Dynamics: A Toy Model

We construct a quantum measure on the power set of non-cyclic oriented graphs of N points, drawing inspiration from 1-dimensional directed percolation. Quantum interference patterns lead to properties which do not appear to have any analogue in classical percolation. Most notably, instead of the single phase transition of classical percolation, the quantum model displays two distinct crossover points. Between these two points, spacetime questions such as "does the network percolate" have no definite or probabilistic answer.Comment: 28 pages incl. 5 figure

Percolation and Connectivity on the Signal to Interference Ratio Graph

A wireless communication network is considered where any two nodes are connected if the signal-to-interference ratio (SIR) between them is greater than a threshold. Assuming that the nodes of the wireless network are distributed as a Poisson point process (PPP), percolation (unbounded connected cluster) on the resulting SIR graph is studied as a function of the density of the PPP. For both the path-loss as well as path-loss plus fading model of signal propagation, it is shown that for a small enough threshold, there exists a closed interval of densities for which percolation happens with non-zero probability. Conversely, for the path-loss model of signal propagation, it is shown that for a large enough threshold, there exists a closed interval of densities for which the probability of percolation is zero. Restricting all nodes to lie in an unit square, connectivity properties of the SIR graph are also studied. Assigning separate frequency bands or time-slots proportional to the logarithm of the number of nodes to different nodes for transmission/reception is sufficient to guarantee connectivity in the SIR graph.Comment: To appear in the Proceedings of the IEEE Conference on Computer Communications (INFOCOM 2012), to be held in Orlando Florida Mar. 201

Connectivity in Sub-Poisson Networks

We consider a class of point processes (pp), which we call {\em sub-Poisson}; these are pp that can be directionally-convexly ($dcx$) dominated by some Poisson pp. The $dcx$ order has already been shown useful in comparing various point process characteristics, including Ripley's and correlation functions as well as shot-noise fields generated by pp, indicating in particular that smaller in the $dcx$ order processes exhibit more regularity (less clustering, less voids) in the repartition of their points. Using these results, in this paper we study the impact of the $dcx$ ordering of pp on the properties of two continuum percolation models, which have been proposed in the literature to address macroscopic connectivity properties of large wireless networks. As the first main result of this paper, we extend the classical result on the existence of phase transition in the percolation of the Gilbert's graph (called also the Boolean model), generated by a homogeneous Poisson pp, to the class of homogeneous sub-Poisson pp. We also extend a recent result of the same nature for the SINR graph, to sub-Poisson pp. Finally, as examples we show that the so-called perturbed lattices are sub-Poisson. More generally, perturbed lattices provide some spectrum of models that ranges from periodic grids, usually considered in cellular network context, to Poisson ad-hoc networks, and to various more clustered pp including some doubly stochastic Poisson ones.Comment: 8 pages, 10 figures, to appear in Proc. of Allerton 2010. For an extended version see http://hal.inria.fr/inria-00497707 version

Spatial networks with wireless applications

Many networks have nodes located in physical space, with links more common between closely spaced pairs of nodes. For example, the nodes could be wireless devices and links communication channels in a wireless mesh network. We describe recent work involving such networks, considering effects due to the geometry (convex,non-convex, and fractal), node distribution, distance-dependent link probability, mobility, directivity and interference.Comment: Review article- an amended version with a new title from the origina