Suppose that ball-shaped sensors wander in a bounded domain. A sensor does not know its location but does know when it overlaps a nearby sensor. We say that an evasion path exists in this sensor network if a moving intruder can avoid detection. In 'Coordinate-free coverage in sensor networks with controlled boundaries via homology', Vin de Silva and Robert Ghrist give a necessary condition, depending only on the time-varying connectivity data of the sensors, for an evasion path to exist. Using zigzag persistent homology, we provide an equivalent condition that moreover can be computed in a streaming fashion. However, no method with time-varying connectivity data as input can give necessary and sufficient conditions for the existence of an evasion path. Indeed, we show that the existence of an evasion path depends not only on the fibrewise homotopy type of the region covered by sensors but also on its embedding in spacetime. For planar sensors that also measure weak rotation and distance information, we provide necessary and sufficient conditions for the existence of an evasion path.
Bibliographical noteFunding Information:
This work was supported by the National Science Foundation (grant numbers DMS 0905823 and DMS 0964242), the Air Force Office of Scientific Research (grant numbers FA9550-09-1-643 and FA9550-09-1-0531), and the National Institutes of Health (grant number I-U54-ca49145-01). H. Adams was supported by a Ric Weiland Graduate Fellowship at Stanford University and by the Institute for Mathematics and its Applications.
© The Author(s) 2014.
- Mobile sensor networks
- fibrewise embeddings
- zigzag persistence