## Network delay inference from additive metrics, Preprint. Available at Arxiv: math.PR/0604367 (2006)

Citations: | 7 - 1 self |

### BibTeX

@MISC{Bhamidi06networkdelay,

author = {Shankar Bhamidi and Ram Rajagopal and Sébastien Roch},

title = {Network delay inference from additive metrics, Preprint. Available at Arxiv: math.PR/0604367},

year = {2006}

}

### OpenURL

### Abstract

We use computational phylogenetic techniques to solve a central problem in inferential network monitoring. More precisely, we design a novel algorithm for multicast-based delay inference, that is, the problem of reconstructing delay characteristics of a network from end-to-end delay measurements on network paths. Our inference algorithm is based on additive metric techniques used in phylogenetics. It runs in polynomial time and requires a sample of size only poly(log n). We also show how to recover the topology of the routing tree. 1

### Citations

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Citation Context ...e [11, 25] for details. 62.2 Distorted Metric Algorithms Classical distance-based reconstruction algorithms (that is, those methods based on tree metrics) such as UPGMA [26] or Neighbor-Joining (NJ) =-=[24]-=-, typically make use of all pairwise distances between leaves. This leads to difficulties because “long” distances are more “noisy” and require a large number of samples to be accurately estimated. Fo... |

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Citation Context ...topology of the routing tree as well as to estimate link properties of this tree. The main link property we consider here is the delay distribution. The multicast inference approach was introduced in =-=[4, 23]-=-. A core difficulty of the problem is to devise efficient, scalable algorithms which consistently estimate the desired network properties. Several techniques have been used in the network tomography l... |

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Citation Context ...res a sample of size only poly(log n). We also show how to recover the topology of the routing tree. 1 Introduction Network tomography. Inferential network monitoring—also known as network tomography =-=[27]-=-— consists in reconstructing various properties of large communication networks from indirect measurements in order to facilitate the management of these networks. Network inference can be achieved by... |

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Citation Context ...rom the function W, assuming further that we are given the tree T. For this purpose, we use a standard algorithm from combinatorial phylogenetics—related to the so-called Four-Point Method of Buneman =-=[3]-=- (see also [11, 25]). We will refer to this algorithm as the Additive Function Inference (AFI) algorithm. See Figures 2 and 3. Algorithm Additive Function Inference Input: tree T, function W at the le... |

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Citation Context ...y from the data at the leaves. As it turns out, even moments are rather straightforward to estimate inductively while odd moments are trickier. Also, as in the tree reconstruction algorithm (see also =-=[9, 17]-=-), the AFI algorithm uses only “short” paths during the estimation process, which allows a significant reduction in the sample size (see Propositions 1, 2 and Theorems 4, 5 for details). 4.1 Additive ... |

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Citation Context ...mate the desired network properties. Several techniques have been used in the network tomography literature, notably maximum pseudo-likelihood, EM algorithms and Markov chain Monte Carlo methods. See =-=[6]-=- for a detailed survey and bibliographic references. In this paper, we introduce a new methodology for multicast delay inference inspired by techniques from the field of phylogenetics in biology, that... |

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Citation Context ... a consistent estimator of the routing tree, but no convergence rate is given. Note, however, that RNJ has in fact a high sample complexity due to its reliance on the diameter of the tree. See, e.g., =-=[1]-=-. See also our discussion about diameter v. depth in Section 2.2. Here, we make use of state-of-art phylogenetic reconstruction techniques to derive a low sample complexity algorithm for routing tree ... |

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Citation Context ... joint distributions on pairs of leaves whereas the reconstruction of delays (in the asymmetric case) required the joint distributions on triples of leaves. A similar situation holds in phylogenetics =-=[7]-=-. It could be interesting to prove that this is indeed necessary in some sense. 4. Throughout, the model was assumed to be static. In real-life networks, characteristics of the network change over tim... |

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Citation Context ...s of internal degrees higher than 3 by introducing a variant of NJ called Rooted Neighbor-Joining (RNJ) [21] (based on a technique equivalent to what is known in phylogenetics as the Farris transform =-=[10]-=-). They show more precisely that RNJ is a consistent estimator of the routing tree, but no convergence rate is given. Note, however, that RNJ has in fact a high sample complexity due to its reliance o... |

26 | S.: Learning nonsingular phylogenies and hidden markov models
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Citation Context ...y from the data at the leaves. As it turns out, even moments are rather straightforward to estimate inductively while odd moments are trickier. Also, as in the tree reconstruction algorithm (see also =-=[9, 17]-=-), the AFI algorithm uses only “short” paths during the estimation process, which allows a significant reduction in the sample size (see Propositions 1, 2 and Theorems 4, 5 for details). 4.1 Additive ... |

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Citation Context ...he following. From measurement of sequences of molecular data at the leaves, one seeks to reconstruct the topology of the evolutionary tree as well as mutation characteristics along the branches. See =-=[11]-=- and [25] for an overview of the field of phylogenetics. Various statistical and computational techniques have been used to solve the phylogenetic reconstruction problem: maximum likelihood, bayesian,... |

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Citation Context .... We first need the following definition which formalizes the idea that short distances are accurately estimated (and that long distances can in some sense be ignored). Definition 4 (Distorted Metric =-=[16, 12]-=-) Let T = (V,E) be a tree with leaf set L and edge weight function w : E → R++. Let W : L × L → R+ be the corresponding tree metric. Fix ˜τ, ˜ M > 0. We say that ̂ W : L × L → (0,+∞] is a (˜τ, ˜ M)-di... |

13 | On the Complexity of Distance-based Evolutionary Tree Reconstruction
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Citation Context .... We first need the following definition which formalizes the idea that short distances are accurately estimated (and that long distances can in some sense be ignored). Definition 4 (Distorted Metric =-=[16, 12]-=-) Let T = (V,E) be a tree with leaf set L and edge weight function w : E → R++. Let W : L × L → R+ be the corresponding tree metric. Fix ˜τ, ˜ M > 0. We say that ̂ W : L × L → (0,+∞] is a (˜τ, ˜ M)-di... |

9 | S (2009) Phylogenies without branch bounds: Contracting the short, pruning the deep
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Citation Context ...Section 3. We then present and analyze our delay inference algorithm in Section 4. 2 Phylogenetic Reconstruction Techniques In this section, we summarize and adapt to our setting the DMR algorithm of =-=[8]-=-. 2.1 Basics We begin with a few basic notions from phylogenetics. Tree metrics. In phylogenetics, the notion of a tree metric is useful for reconstructing the topology of phylogenies. We use the nota... |

9 |
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Citation Context ...ely estimated. For instance, in the phylogenetic context, the widely used NJ algorithm is computationally efficient, but it is known to require exponentially many samples—even for simple linear trees =-=[13]-=-. An important breakthrough was made in [9] where it was shown that it was in fact enough to use “short” distances to fully recover the tree under reasonable assumptions. To help understand this resul... |

7 | A markov random field approach to multicast-based network inference problems
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Citation Context ...gorithms with low sample complexity are essential. Concurrently to our work, Liang et al. [15] used similar ideas to tackle the related multicast packet loss inference problem. Also, Ni and Tatikonda =-=[19]-=- independently proposed a Markov-based inference algorithm similar to ours for multicast delay inference—although our work appears to be the first rigorous analysis of the sample complexity of this ap... |

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Citation Context ...Throughout, the model was assumed to be static. In real-life networks, characteristics of the network change over time. One could try to adapt our algorithm to a more dynamic setting. See for example =-=[5]-=- for a discussion of temporal issues. 20Acknowledgments We thank Gang Liang, Elchanan Mossel, and Bin Yu for discussions and encouragements. S.R. gratefully acknowledges the partial support of CIPRES... |

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Citation Context ...of all of its bipartitions. See [11, 25] for details. 62.2 Distorted Metric Algorithms Classical distance-based reconstruction algorithms (that is, those methods based on tree metrics) such as UPGMA =-=[26]-=- or Neighbor-Joining (NJ) [24], typically make use of all pairwise distances between leaves. This leads to difficulties because “long” distances are more “noisy” and require a large number of samples ... |

4 | Network Routing Topology Inference from End-to-End Measurements
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(Show Context)
Citation Context ... characteristics. More explicit formulas are given in the limit of small delays. The algorithm is tested on small networks and the dependence on the size is not given. More recently, Ni and Tatikonda =-=[19, 20, 21, 22]-=-—in work subsequent to ours [2]—used phylogenetic techniques to recover the routing tree topology in this context. Similarly to the current paper, they use distance-based techniques. The basic algorit... |

3 |
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Citation Context ...n nature. Typical networks undergo sporadic mediumto large-scale changes in structure over time, therefore algorithms with low sample complexity are essential. Concurrently to our work, Liang et al. =-=[15]-=- used similar ideas to tackle the related multicast packet loss inference problem. Also, Ni and Tatikonda [19] independently proposed a Markov-based inference algorithm similar to ours for multicast d... |

1 |
link parameter estimators based on end-to-end measurements
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(Show Context)
Citation Context ... characteristics. More explicit formulas are given in the limit of small delays. The algorithm is tested on small networks and the dependence on the size is not given. More recently, Ni and Tatikonda =-=[19, 20, 21, 22]-=-—in work subsequent to ours [2]—used phylogenetic techniques to recover the routing tree topology in this context. Similarly to the current paper, they use distance-based techniques. The basic algorit... |

1 |
tomography based on additive metrics
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(Show Context)
Citation Context ... characteristics. More explicit formulas are given in the limit of small delays. The algorithm is tested on small networks and the dependence on the size is not given. More recently, Ni and Tatikonda =-=[19, 20, 21, 22]-=-—in work subsequent to ours [2]—used phylogenetic techniques to recover the routing tree topology in this context. Similarly to the current paper, they use distance-based techniques. The basic algorit... |