Mobile Geometric Graphs: Detection, Coverage and Percolation
Abstract
We consider the following dynamic Boolean model introduced by van den Berg, Meester and White (1997). At time 0, let the nodes of the graph be a Poisson point process in R^d 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 if their distance is at most r. We study three features in this model: detection (the time until a target pointfixed or movingis within distance r from 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 features by combining ideas from stochastic geometry, coupling and multiscale analysis.
 Publication:

arXiv eprints
 Pub Date:
 July 2010
 DOI:
 10.48550/arXiv.1008.0075
 arXiv:
 arXiv:1008.0075
 Bibcode:
 2010arXiv1008.0075P
 Keywords:

 Mathematics  Probability;
 Computer Science  Discrete Mathematics