Failure Filtrations for Fenced Sensor Networks

In this paper we consider the question of sensor network coverage for a 2-dimensional domain. We seek to compute the probability that a set of sensors fails to cover given only non-metric, local (who is talking to whom) information and a probability distribution of failure of each node. This builds on the work of de Silva and Ghrist who analyzed this problem in the deterministic situation. We first show that a it is part of a slightly larger class of problems which is #P-complete, and thus fast algorithms likely do not exist unless P$=$NP. We then give a deterministic algorithm which is feasible in the case of a small set of sensors, and give a dynamic algorithm for an arbitrary set of sensors failing over time which utilizes a new criterion for coverage based on the one proposed by de Silva and Ghrist. These algorithms build on the theory of topological persistence.

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