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Finding the Right Tree: Topology Inference Despite Spatial Dependences

Rhys Bowden, Darryl Veitch

2018
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IEEE Transactions on Information Theory
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Network tomographic techniques have almost exclusively been built on a strong assumption of mutual independence of link processes. We introduce model classes for link loss processes with non-trivial spatial dependencies, for which the tree topology is nonetheless identifiable from leaf measurements using multicast probing. We show that these classes are large in a well defined sense, and we provide an algorithm, SLTD2, capable of returning the correct topology with certainty in the limit of
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... nite data. Index Terms Network tomography, spatial dependence, identifiability, topology inference, loss measurement, multicast-trees. I. INTRODUCTION N ETWORK Tomography is concerned with inferring underlying details of networks from incomplete or inaccurate measurements. In network link tomography over trees, a unique sender node sends many test packets, or probes, to m > 1 receiver nodes. The design of network routing is such that, typically, the sequences of network links traversed by such probes will naturally possess a tree topology, a measurement tree subgraph of the underlying network graph. Each receiver records measurements from the probes that arrive to it, and these are used to infer conditions at links or nodes inside the tree. Particular attention has been paid to multicast-trees. In multicasting, a packet sent from a source toward m destinations is duplicated only when needed, at branch points, rather than the source sending m copies. In multicast-tree tomography this functionality is abstracted into the powerful property of perfectly shared-history of receiver probes above branch points. Multicast-tree tomography can be seen both as an exploitation of IP protocol functionality for measurement ends [1]-[9], or as a means to infer properties of a multicast-tree to improve the performance of applications using it [10] . It can also be viewed as a simplifying abstraction to explore the limitations of tree-based tomography in general, in particular because unicast approaches exist which can emulate the shared-history property of multicast. Tree-based tomography has considered inference of link loss processes (e.g. [2], [11])), link delays (e.g. [4], [6])), and tree topology (e.g. [7], [10], [12], [13]). In almost every case full spatial independence, that is the mutual independence of all loss or delay processes across different links, was a core assumption underlying the inference. We refer to any model incorporating this classical assumption as a classical model. Two works where non-classical models are considered are those of Caceras et al. [2] and Ghita et al. [14] , both in the context of loss tomography. We discuss these in Section II. This paper describes loss-based topology inference in multicast-trees, in the context of non-trivial spatial dependence of link loss processes. In order to focus on fundamental issues of identifiability, we work in the ideal context of 'infinite data', meaning we have access to the true m dimensional joint distribution of loss measurements across all leaves/receivers. Our first contribution lies in the definition of new model classes with non-trivial spatial dependencies. The significance of these classes is that they each possess the non-trivial property of topological determinism, which implies that the topology of each model within the class is identifiable, even if the full model itself is not. We begin by defining a natural generalization of classical models, the Classically Equivalent (CE) class. We then introduce a more explicit and richer class, the Jump Independent (JI) models, which is physically meaningful in terms of real networks, and from it define two related classes, the Agreeable JI models (AJI), and the Agreeable JI Equivalent models (AJIE), which are topologically determinate. We show that each is large in the sense of the dimensionality of its parameter space, and much larger than the classical class, and also that, although their topologies are identifiable, the loss processes on links are not. Figure 6 shows the relationship between these classes. Our second contribution is in the definition of topology recovery algorithms and proofs of their correctness. We first describe Shared Loss Topology Discovery (SLTD), which is capable of fully identifying the topology of CE models without error. This SLTD algorithm was inspired by, and is closely related to, that proposed in [7], [10] for topology inference under spatial independence. We show how, in the infinite data case, it is based on a property we call 'certain paternity' which is powerful enough to allow SLTD to be applied much more widely. We then describe another algorithm, SLTD2, designed to recover the topology of models from the AJI class, a far more challenging task. It is also based on shared loss between receiver pairs, but using a different and more general certain paternity property. Using an approach of broad applicability, we prove that SLTD2 also recovers the topology of models in the AJIE class, which includes the CE class as a special case. Hence SLTD2 is a new algorithm that can be used instead of SLTD but with much wider applicability. Importantly, it is based on the joint distributions of pairs of receivers only, yet captures

doi:10.1109/tit.2017.2739779
fatcat:jo2emd3ze5hzvk6zbzhzudd6oe