Tight bounds to localize failure nodes on trees, grids and through embeddings under boolean network tomography

Maximal identifiability was recently introduced in boolean network tomography to measure the maximal number of corrupted nodes which can be uniquely localized in sets of end-to-end measurement paths on networks ([1,2]). We contribute to the study of maximal identifiability proving upper and lower bounds on this measure for sets of end-to-end paths defined on different network topologies. First we show some results relating maximal identifiability to structural graph measures like the minimal degree or the number of nodes of the network connected to external monitors. For trees we show that the maximal identifiability is upper bounded by 1. We define a property (monitor balanced) on the monitor placement (that is deciding what nodes in the graph to link to external monitors) which guarantees on trees a maximal identifiability of 1. We also describe a strategy using a minimal number of monitors to always define a monitor-balanced placement. In search for topologies better than trees from the point of view of failure identifiability we consider the case of grids. We prove that choosing any 4 nodes to link to monitors, maximal identifiability on 2-dimensional grids is at least 1 and at most 2. Moreover we prove that this result is optimal, namely that using less than 8 monitors we cannot always reach an identifiability of 2. We also consider the case of directed and acyclic graph. For directed 2-dimensional grids we define a monitor placement on 4(k−1) nodes obtaining that maximal identifiability is exactly 2. We show that this monitor placement is unique and optimal. Finally we explore how maximal identifiability grows under order-isomorphisms, that is bijective embeddings of directed graphs. Using these results we prove that under the operation of transitive closure maximal identifiability grows linearly