  To the coverage problem of sensor networks, V. de Silva and R. Ghrist (2007)
developed several approaches based on (persistent) homology theory. Their
criteria for the coverage are formulated on the Rips complexes constructed by
the sensors, in which their locations are supposed to be fixed. However, the
sensors are in general affected by perturbations (e.g., natural phenomena), and
hence the stability of the coverage criteria should be also discussed. In this
paper, we present a coverage theorem stable under perturbation. Furthermore, we
also introduce a method of eliminating redundant cover after perturbation. The
coverage theorem is derived by extending the Rips interleaving theorem studied
by F. Chazal, V. de Silva, and S. Oudot (2013) into an appropriate relative
version.
