Graph reconstruction from path correlation data

Abstract

A communication network can be modeled as a directed connected graph with edge weights that characterize performance metrics such as loss and delay. Network tomography aims to infer these edge weights from their pathwise versions measured on a set of intersecting paths between a subset of boundary vertices, and even the underlying graph when this is not known. In particular, temporal correlations between path metrics have been used infer composite weights on the subpath formed by the path intersection. We call these subpath weights the Path Correlation Data.In this paper we ask the following question: when can the underlying weighted graph be recovered knowing only the boundary vertices and the Path Correlation Data? We establish necessary and sufficient conditions for a graph to be reconstructible from this information, and describe an algorithm to perform the reconstruction. Subject to our conditions, the result applies to directed graphs with asymmetric edge weights, and accommodates paths arising from asymmetric routing in the underlying communication network. We also describe the relationship between the graph produced by our algorithm and the true graph in the case that our conditions are not satisfied. out to be rather natural -for the reconstruction of a graph from its PCD and present an algorithm to achieve it. In the case when the underlying graph violates the reconstructibility conditions, we describe the result of the algorithm -it turns out to be the "simplest" routing network that produces the observed PCD.Network Tomography and the Inversion Problem. The form of our model and the assumptions we make originate from a body of work developed under the term Network Tomography [19] that seeks to infer link metrics and even the underlying network topology from the measured metric values on paths traversing the network between a set of routers at the network boundary, represented by V B . This setting is similar to other graph reconstruction problems, such as tomography of electrical resistance networks (see, e.g., [8,6]), optical networks [10], and graph reconstruction from Schrödinger-type spectral data (see, e.g., [3,4]). However, in a communication network model there is a single path between given origin and destination, in contrast to the electrical current flowing between two points a resistive medium via all possible paths. In this sense, our model is more similar to combinatorial reconstruction problems [1,15].In many practical cases the communication network metrics are additive in the sense that the sum of metric values over links in a path corresponds to the same performance metric for the path. Examples of additive metrics include mean packet delay, log packet transmission probability, and variances in an independent link metric model. For additive metrics, a putative solution to the network tomography problem attempts to invert a linear relation expression between the set of path metrics D and the link metrics W in the formHere A is the incidence matrix of links over paths, A P, is 1 if path...

Optimizing Consistent Merging and Pruning of Subgraphs in Network Tomography

“…Third: how are these algorithms best adapted to work with finite measurement data in the sense of being applicable and performing accurately? The first question has recently been answered for a wide class of inference problems on networks with asymmetric paths between host pairs [3]. Network level inference is performed by fusing source and destination based trees at each measurement host, characterized by their Path Correlation Data (PCD), namely the weight of the intersection of any two paths that share an origin or destination.…”

“…Network level inference is performed by fusing source and destination based trees at each measurement host, characterized by their Path Correlation Data (PCD), namely the weight of the intersection of any two paths that share an origin or destination. Necessary and sufficient conditions for a network graph to be reconstructible from the PCD were established in [3] together with an explicit reconstruction algorithm. Thus when the PCD are identifiable from path pair measurements, the full network is identifiable under the reconstruction conditions.…”

“…(ii) no interior boundaries: no v ∈ V B is an interior node of any path in P. A partial network without this property can be made conformant by replacing such v by an interior node to which it is connected with a zero weight edge; see e.g. Figure 5 in [3]). The performance metrics are considered additive in the sense that the performance metric of the path P u,v is the sum W u,v = e∈Pu,v w e .…”

“…Difficulties in Merging Weighted Graphs. Lack of consistent path weights prevents application of fusing algorithms that require them [3]. To illustrate this, consider in Figure 1 two weighted simple binary trees T S b 1 and T R b 2 that have been inferred by primitive inference using distinct packet time series involving the same three boundary vertices b 1 , b 2 and b 3 with central vertices c and c respectively.…”

“…To merge, the central vertices c from T S b 1 and c from T R b 2 are inserted in the path joining b 1 and b 2 according to the edge weights in their respective topologies; see e.g. [3]. This results in a directed edge (c, c ) of weight δ = w 1 − v 1 = v 2 − w 2 .…”