Vertex-Connectivity Measures for Node Failure Identification in Boolean Network Tomography
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We investigate three questions in Boolean Network Tomography, related to maximal vertex identifiability, i.e. the maximal number of failing nodes simultaneously identifiable in a network. First, how to characterize the identifiability of the network through structural measures of its topology; second, how many monitors and where to place them to maximize identifiability of failures; third, which tradeoffs there are between the number of monitors and the maximal number of identifiable failures. We first consider Line-of-Sight networks and we characterize the maximal identifiability of such networks highlighting that vertex-connectivity plays a central role. Motivated by this observation, we give a precise characterization of the maximal identifiability in terms of vertex-connectivity for any network: using Menger's theorem, we prove that maximal identifiability is contained in a constant of 1 from vertex connectivity. A consequence of this is a first algorithm based on the well-known Max-Flow problem to decide where to place the monitors in any network in order to maximize identifiability. Finally we initiate the study of maximal identifiability for random networks on two models: Erdös-Rènyi model and Random Regular graphs. The framework proposed in the paper allows a probabilistic analysis of the identifiability in random networks giving a tradeoff between the number of monitors to place and the maximal identifiability.
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