Geometric scale-free random graphs on mobile vertices: broadcast and percolation times

We study the phenomenon of information propagation on mobile geometric scale-free random graphs, where vertices instantaneously pass on information to all other vertices in the same connected component. The graphs we consider are constructed on a Poisson point process of intensity \(\lambda>0\), and the vertices move over time as simple Brownian motions on either \(\mathbb{R}^d\) or the \(d\)-dimensional torus of volume \(n\), while edges are randomly drawn depending on the locations of the vertices, as well as their a priori assigned marks. This includes mobile versions of the age-dependent random connection model and the soft Boolean model. We show that in the ultrasmall regime of these random graphs, information is broadcast to all vertices on a torus of volume \(n\) in poly-logarithmic time and that on \(\mathbb{R}^d\), the information will reach the infinite component before time \(t\) with stretched exponentially high probability, for any \(\lambda>0\).