We study maximal identifiability, a measure recently introduced in Boolean Network Tomography to characterize networks' capability to localize failure nodes in end-to-end path measurements. We prove tight upper and lower bounds on the maximal identifiability of failure nodes for specific classes of network topologies, such as trees and d-dimensional grids, in both directed and undirected cases. We prove that directed d-dimensional grids with support n have maximal identifiability d using 2d(n − 1) + 2 monitors; and in the undirected case we show that 2d monitors suffice to get identifiability of d − 1. We then study identifiability under embeddings: we establish relations between maximal identifiability, embeddability and graph dimension when network topologies are modeled as DAGs. Our results suggest the design of networks over N nodes with maximal identifiability Ω(log N ) using O(log N ) monitors and a heuristic to boost maximal identifiability on a given network by simulating d-dimensional grids. We provide positive evidence of this heuristic through data extracted by exact computation of maximal identifiability on examples of small real networks. * A preliminary version of this paper appeared in [11].

In this paper we study the node failure identification problem in undirected graphs by means of Boolean Network Tomography. We argue that vertex connectivity plays a central role. We show tight bounds on the maximal identifiability in a particular class of graphs, the Line of Sight networks. We prove slightly weaker bounds on arbitrary networks. Finally we initiate the study of maximal identifiability in random networks. We focus on two models: the classical Erdős-Rényi model, and that of 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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