Evasion Paths in Mobile Sensor Networks

Suppose that ball-shaped sensors wander in a bounded domain. A sensor doesn't 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 evader 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 graph of the sensors, for an evasion path to exist. Can we sharpen this result? 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 the ambient isotopy class of 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. Furthermore, we study the entire space of evasion paths using a homotopy spectral sequence for diagrams of spaces due to Emmanuel Dror Farjoun and Alexander Zabrodsky.