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

TL;DR

This paper establishes tight bounds on the maximal identifiability of failure nodes in Boolean Network Tomography for specific network topologies, providing theoretical insights and practical heuristics for network design.

Contribution

It derives tight bounds for maximal identifiability in various network classes and introduces a heuristic to enhance identifiability in real networks.

Findings

Directed d-dimensional grids have maximal identifiability d with 2d(n-1)+2 monitors.

Undirected d-dimensional grids require 2d monitors for identifiability of d-1.

A heuristic can improve maximal identifiability in real networks, supported by data from small network examples.

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 model as DAGs. Our results suggest the design of networks over $N$ nodes with maximal identifiability $\Omega(\log N)$ 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.