  Motivated by Krioukov et al.'s model of random hyperbolic graphs for
real-world networks, and inspired by the analysis of a dynamic model of graphs
in Euclidean space by Peres et al., we introduce a dynamic model of hyperbolic
graphs in which vertices are allowed to move according to a Brownian motion
maintaining the distribution of vertices in hyperbolic space invariant. For
different parameters of the speed of angular and radial motion, we analyze tail
bounds for detection times of a fixed target and obtain a complete picture, for
very different regimes, of how and when the target is detected: as a function
of the time passed, we characterize the subset of the hyperbolic space where
particles typically detecting the target are initially located.
  We overcome several substantial technical difficulties not present in
Euclidean space, and provide a complete picture on tail bounds. On the way, we
obtain also new results for the time more general continuous processes with
drift and reflecting barrier spent in certain regions, and we also obtain
improved bounds for independent sums of Pareto random variables.
