TY - JOUR
T1 - Mobile geometric graphs
T2 - Detection, coverage and percolation
AU - Peres, Yuval
AU - Sinclair, Alistair
AU - Sousi, Perla
AU - Stauffer, Alexandre
PY - 2013/6
Y1 - 2013/6
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=84878112315&partnerID=8YFLogxK
UR - http://dx.doi.org/10.1007/s00440-012-0428-1
U2 - 10.1007/s00440-012-0428-1
DO - 10.1007/s00440-012-0428-1
M3 - Article
SN - 0178-8051
VL - 156
SP - 273
EP - 305
JO - Probability Theory and Related Fields
JF - Probability Theory and Related Fields
IS - 1-2
ER -