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

Percolation in Multi-hop Wireless Networks

To be adde

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

An Analytical Expression for k-connectivity of Wireless Ad Hoc Networks

Over the last few years coverage and connectivity of wireless ad hoc networks have fascinated considerable attention. The presented paper analyses and investigates the issues of k-connectivity probability and its robustness in wireless ad hoc-network while considering fading techniques like lognormal fading, Rayleigh fading, and nakagami fading in the ad hoc communication environment, by means of shadowing and fading phenomenon. In case of k-connected wireless sensor network (WSNs), this technique permits the routing of data packets or messages via individual (one or more) of minimum k node disjoint communication paths, but the other remaining paths can also be used. The major contribution of the paper is mathematical expressions for k-connectivity probability

A Framework for Analysis of Connectivity and Performance Bounds in Ad Hoc Networks and Its Application to a Slotted-ALOHA Scenario

In this paper, a framework is proposed to analyse the problems of connectivity and performance in ad-hoc networks through an analytical approach. To this aim, available results regarding the application of percolation theory to the study of connectivity in ad-hoc networks are exploited jointly with communication theory models in order to derive the configuration of network parameters that ensures long range connectivity among nodes and the corresponding available capacity on the wireless medium. The framework is then applied to a slotted ALOHA ad-hoc network. Theoretical and numerical results validate the approach and allow the derivation of interesting design principles for ad-hoc networks that consider the impact of physical and MAC-level parameters on network connectivity and end-to-end performance

Optimal Paths on the Space-Time SINR Random Graph

We analyze a class of Signal-to-Interference-and-Noise-Ratio (SINR) random graphs. These random graphs arise in the modeling packet transmissions in wireless networks. In contrast to previous studies on the SINR graphs, we consider both a space and a time dimension. The spatial aspect originates from the random locations of the network nodes in the Euclidean plane. The time aspect stems from the random transmission policy followed by each network node and from the time variations of the wireless channel characteristics. The combination of these random space and time aspects leads to fluctuations of the SINR experienced by the wireless channels, which in turn determine the progression of packets in space and time in such a network. This paper studies optimal paths in such wireless networks in terms of first passage percolation on this random graph. We establish both "positive" and "negative" results on the associated time constant. The latter determines the asymptotics of the minimum delay required by a packet to progress from a source node to a destination node when the Euclidean distance between the two tends to infinity. The main negative result states that this time constant is infinite on the random graph associated with a Poisson point process under natural assumptions on the wireless channels. The main positive result states that when adding a periodic node infrastructure of arbitrarily small intensity to the Poisson point process, the time constant is positive and finite

A Unifying Framework for Local Throughput in Wireless Networks

With the increased competition for the electromagnetic spectrum, it is important to characterize the impact of interference in the performance of a wireless network, which is traditionally measured by its throughput. This paper presents a unifying framework for characterizing the local throughput in wireless networks. We first analyze the throughput of a probe link from a connectivity perspective, in which a packet is successfully received if it does not collide with other packets from nodes within its reach (called the audible interferers). We then characterize the throughput from a signal-to-interference-plus-noise ratio (SINR) perspective, in which a packet is successfully received if the SINR exceeds some threshold, considering the interference from all emitting nodes in the network. Our main contribution is to generalize and unify various results scattered throughout the literature. In particular, the proposed framework encompasses arbitrary wireless propagation effects (e.g, Nakagami-m fading, Rician fading, or log-normal shadowing), as well as arbitrary traffic patterns (e.g., slotted-synchronous, slotted-asynchronous, or exponential-interarrivals traffic), allowing us to draw more general conclusions about network performance than previously available in the literature.Comment: Submitted for journal publicatio