Fundamental limits of failure identifiability by Boolean Network Tomography

Boolean network tomography is a powerful tool to infer the state (working/failed) of individual nodes from path-level measurements obtained by egde-nodes. We consider the problem of optimizing the capability of identifying network failures through the design of monitoring schemes. Finding an optimal solution is NP-hard and a large body of work has been devoted to heuristic approaches providing lower bounds. Unlike previous works, we provide upper bounds on the maximum number of identifiable nodes, given the number of monitoring paths and different constraints on the network topology, the routing scheme, and the maximum path length. The proposed upper bounds represent a fundamental limit on the identifiability of failures via Boolean network tomography. This analysis provides insights on how to design topologies and related monitoring schemes to achieve the maximum identifiability under various network settings. Through analysis and experiments we demonstrate the tightness of the bounds and efficacy of the design insights for engineered as well as real network

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

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

The Fault-Finding Capacity of the Cable Network When Measured Along Complete Paths

We look into whether or not it is possible to find the exact location of a broken node in a communication network by using the binary state (normal or failed) of each link in the chain. To find out where failures are in a group of nodes of interest, it is necessary to link the different states of the routes to the different failures at the nodes. Due to the large number of possible node failures that need to be listed, it may be hard to check this condition on large networks. The first important thing we've added is a set of criteria that are both enough and necessary for testing in polynomial time whether or not a set of nodes has a limited number of failures. As part of our requirements, we take into account not only the architecture of the network but also the positioning of the monitors. We look at three different types of probing methods. Each one is different depending on the nature of the measurement paths, which can be random, controlled but not cycle-free, or uncontrolled (depending on the default routing protocol). Our second contribution is an analysis of the greatest number of failures (anywhere in the network) for which failures within a particular node set can be uniquely localized and the largest node set within which failures can be uniquely localized under a given constraint on the overall number of failures in the network. Both of these results are based on the fact that failures can be uniquely localized only if there is a constraint on the overall number of failures. When translated into functions of a per-node attribute, the sufficient and necessary conditions that came before them make it possible for an efficient calculation of both measurements

Vertex-Connectivity for Node Failure Identification in Boolean Network Tomography

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

Node Failure Localization via Network Tomography

We investigate the problem of localizing node failures in a communication network from end-to-end path measure-ments, under the assumption that a path behaves normally if and only if it does not contain any failed nodes. To uniquely localize node failures, the measurement paths must show dif-ferent symptoms under different failure events, i.e., for any two distinct sets of failed nodes, there must be a measure-ment path traversing one and only one of them. This condi-tion is, however, impractical to test for large networks. Our first contribution is a characterization of this condition in terms of easily verifiable conditions on the network topol-ogy with given monitor placements under three families of probing mechanisms, which differ in whether measurement paths are (i) arbitrarily controllable, (ii) controllable but cycle-free, or (iii) uncontrollable (i.e., determined by the de-fault routing protocol). Our second contribution is a char-acterization of the maximum identifiability of node failures, measured by the maximum number of simultaneous failures that can always be uniquely localized. Specifically, we bound the maximal identifiability from both the upper and the lower bounds which differ by at most one, and show that these bounds can be evaluated in polynomial time. Finally, we quantify the impact of the probing mechanism on the capability of node failure localization under different prob-ing mechanisms on both random and real network topolo-gies. We observe that despite a higher implementation cost, probing along controllable paths can significantly improve a network’s capability to localize simultaneous node failures

Active Topology Inference using Network Coding

Our goal is to infer the topology of a network when (i) we can send probes between sources and receivers at the edge of the network and (ii) intermediate nodes can perform simple network coding operations, i.e., additions. Our key intuition is that network coding introduces topology-dependent correlation in the observations at the receivers, which can be exploited to infer the topology. For undirected tree topologies, we design hierarchical clustering algorithms, building on our prior work. For directed acyclic graphs (DAGs), first we decompose the topology into a number of two-source, two-receiver (2-by-2) subnetwork components and then we merge these components to reconstruct the topology. Our approach for DAGs builds on prior work on tomography, and improves upon it by employing network coding to accurately distinguish among all different 2-by-2 components. We evaluate our algorithms through simulation of a number of realistic topologies and compare them to active tomographic techniques without network coding. We also make connections between our approach and alternatives, including passive inference, traceroute, and packet marking

A network tomography approach for traffic monitoring in smart cities

Various urban planning and managing activities required by a Smart City are feasible because of traffic monitoring. As such, the thesis proposes a network tomography-based approach that can be applied to road networks to achieve a cost-efficient, flexible, and scalable monitor deployment. Due to the algebraic approach of network tomography, the selection of monitoring intersections can be solved through the use of matrices, with its rows representing paths between two intersections, and its columns representing links in the road network. Because the goal of the algorithm is to provide a cost-efficient, minimum error, and high coverage monitor set, this problem can be translated into an optimization problem over a matroid, which can be solved efficiently by a greedy algorithm. Also as supplementary, the approach is capable of handling noisy measurements and a measurement-to-path matching. The approach proves a low error and a 90% coverage with only 20% nodes selected as monitors in a downtown San Francisco, CA topology --Abstract, page iv