Tight Bounds for Maximal Identifiability of Failure Nodes in Boolean Network Tomography

Abstract

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].

A network tomography approach for anomaly localization in Service Function Chaining

“…The proposed solution gives some insights for the design of monitoring strategies that maximize the number of identifiable nodes under different probing scenarios. Finally, let us cite [9] that studied the impact of topology properties on the maximal identifiability. They establish a relation between the different typologies classes (directed, undirected, trees, d-dimensional grids, and bounded-degree graphs) and the identifiability of Boolean metrics.…”

Counting and localizing defective nodes by Boolean network tomography

“…The condition introduced as k-identifiability (for P) states that any two distinct node sets of size at most k can be separated by at least a path in P. k-identifiability initially introduced for link failure detection [10,14], was later studied with success also for node failure detection [2,6,8,[11][12][13]. If this condition is true for a set of measurements paths P it ensures that if there are at most k failure nodes in P then these nodes can be identified unambiguously.…”

“…In the work [6,7] we studied k-identifiability from the topological point of view of the graph underlying P. We were proving tight bounds on the maximum identifiability that can be reached in the case of topologies like trees, grids and hypergrids and under embeddings on directed graphs. Our results culminated with a heuristic to design networks with a high degree of identifiability or to modify a network to boost identifiability.…”

“…Our results culminated with a heuristic to design networks with a high degree of identifiability or to modify a network to boost identifiability. In the work [8] we were employing Menger's theorem establishing a precise relation of µ(P) with the vertex connectivity of the graph underlying P. We generalize results in [6] to Line-of-Sight Networks and started the study of identifiability conditions on random graphs and random regular graphs.…”

“…Duffield, who as first introduced boolean network tomography [3] to identify network failure components, proposed an inference algorithm based on BNT to identify sets of failure links. The BNT approach was later studied also to identify node failures in networks [2,6,8,[11][12][13].…”