  Static wireless networks are by now quite well understood mathematically
through the random geometric graph model. By contrast, there are relatively few
rigorous results on the practically important case of mobile networks, in which
the nodes move over time; moreover, these results often make unrealistic
assumptions about node mobility such as the ability to make very large jumps.
In this paper we consider a realistic model for mobile wireless networks which
we call mobile geometric graphs, and which is a natural extension of the random
geometric graph model. We study two fundamental questions in this model:
detection (the time until a given "target" point - which may be either fixed or
moving - is detected by the network), and percolation (the time until a given
node is able to communicate with the giant component of the network). For
detection, we show that the probability that the detection time exceeds t is
\exp(-\Theta(t/\log t)) in two dimensions, and \exp(-\Theta(t)) in three or
more dimensions, under reasonable assumptions about the motion of the target.
For percolation, we show that the probability that the percolation time exceeds
t is \exp(-\Omega(t^\frac{d}{d+2})) in all dimensions d\geq 2. We also give a
sample application of this result by showing that the time required to
broadcast a message through a mobile network with n nodes above the threshold
density for existence of a giant component is O(\log^{1+2/d} n) with high
probability.
