Results on Finite Wireless Networks on a Line

Abstract

Abstract-Today, due to the vast amount of literature on largescale wireless networks, we have a fair understanding of the asymptotic behavior of such networks. However, in real world we have to face finite networks for which the asymptotic results cease to be valid. We refer to networks as being finite when the number of nodes is less than a few hundred. Here we study a model of wireless networks, represented by random geometric graphs. In order to address a wide class of the network's properties, we study the threshold phenomena. Being extensively studied in the asymptotic case, the threshold phenomena occurs when a graph theoretic property (such as connectivity) of the network experiences rapid changes over a specific interval of the underlying parameter. Here, we find an upper bound for the threshold width of finite line networks represented by random geometric graphs. These bounds hold for all monotone properties of such networks. We then turn our attention to an important non-monotone characteristic of line networks which is the medium access (MAC) layer capacity, i.e. the maximum number of possible concurrent transmissions. Towards this goal, we provide an algorithm which finds a maximal set of concurrent non-interfering transmissions and further derive lower and upper bounds for the cardinality of the set. Using simulations, we show that these bounds serve as reasonable estimates for the actual value of the MAC-layer capacity.

Cascading Failures in Load-Dependent Finite-Size Random Geometric Networks

“…When R − r v ≥ R a , A is entirely included in v's neighborhood. Thus, E[N] = λπR 2 a , and E[l u ] is given by (14), together leading to (18).…”

“…Furthermore, we are interested in studying cascading failures in networks with geometric characteristics such as electrical power grids and wireless communication networks, which could be well-modeled as random geometric graphs. Indeed, random geometric graphs have been widely used in studying wireless networks (see [14] and references therein). As expected, it is shown that geometry plays an important role in quantifying the topology of the smart grid communication and control networks [15].…”