We consider the following dynamic Boolean model introduced by van den Berg et al. (Stoch. Process. Appl. 69:247–257, 1997). At time 0, let the nodes of the graph be a Poisson point process in Rd with constant intensity and let each node move independently according to Brownian motion. At any time t, we put an edge between every pair of nodes whose distance is at most r. We study three fundamental problems in this model: detection (the time until a target point—fixed or moving—is within distance r of some node of the graph); coverage (the time until all points inside a finite box are detected by the graph); and percolation (the time until a given node belongs to the infinite connected component of the graph). We obtain precise asymptotics for these quantities by combining ideas from stochastic geometry, coupling and multi-scale analysis.
Peres, Y., Sinclair, A., Sousi, P., & Stauffer, A. (2013). Mobile geometric graphs: Detection, coverage and percolation. Probability Theory and Related Fields, 156(1-2), 273-305. https://doi.org/10.1007/s00440-012-0428-1