On the Spectral Characteristics of Ad Hoc Networks and their Mobility Properties

Random geometric graphs have proven to be extremely useful in modeling static wireless ad-hoc and sensor networks. The study of these graphs and their spectral properties is a very active field with many interesting applications. In this work, we study the spectral properties of these graphs and in particular we prove the lack of spectral gap in random geometric graphs under a wide range of cases. We also present some results on the spectral properties of generalized weighted versions of these graphs. We later focus on mobile geometric graphs or dynamic Boolean models as were introduced by van den Berg, Meester, and White in 1997. These mobile models have been used to model mobile ad-hoc and sensor networks. Our results are aimed at understanding the dynamics of message spreading in a real-world dynamical network. We simulate a message spreading process and present some results related to the total propagation time under different scenarios. We call total propagation time the time it takes for a certain message to reach all the nodes in the network. This message could be a packet, a virus, a rumor in a social network, or something more general. In particular, we study how the metric, the mobility properties, the presence or absence of obstacles and finally the propagation characteristics affect the dynamics of the message spreading.