  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 intruder can avoid
detection. In "Coordinate-free coverage in sensor networks with controlled
boundaries via homology", Vin deSilva 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.
