On suitability of Euclidean embedding of internet hosts

Abstract

In this paper, we investigate the suitability of embedding Internet hosts into a Euclidean space given their pairwise distances (as measured by round-trip time). Using the classical scaling and matrix perturbation theories, we first establish the (sum of the) magnitude of negative eigenvalues of the (doubly-centered, squared) distance matrix as a measure of suitability of Euclidean embedding. We then show that the distance matrix among Internet hosts contains negative eigenvalues of large magnitude, implying that embedding the Internet hosts in a Euclidean space would incur relatively large errors. Motivated by earlier studies, we demonstrate that the inaccuracy of Euclidean embedding is caused by a large degree of triangle inequality violation (TIV) in the Internet distances, which leads to negative eigenvalues of large magnitude. Moreover, we show that the TIVs are likely to occur locally, hence, the distances among these close-by hosts cannot be estimated accurately using a global Euclidean embedding, in addition, increasing the dimension of embedding does not reduce the embedding errors. Based on these insights, we propose a new hybrid model for embedding the network nodes using only a 2-dimensional Euclidean coordinate system and small error adjustment terms. We show that the accuracy of the proposed embedding technique is as good as, if not better, than that of a 7-dimensional Euclidean embedding.

Dynamic landmark triangles: A simple and efficient mechanism for inter-host latency estimation

“…This figure confirms the finding in [16], in that the errors do increase significantly for estimating shorter distances (note the logarithmic scale on Y-axis) for all methods. For example, the median relative error of distance prediction of GNP for real distance between 100 and 1000 milliseconds is just under 10%, but it increases to 68% for real distances below 10ms.…”

“…As future work, we would like to investigate some optimizations to our approach we mentioned throughout the paper, most notably, accounting for triangle shape during dynamic triangle selection and using a replacing the triangle with a single landmark when it is particularly close to one of the end hosts. We would also like to combine our approach with some orthogonal techniques that have been proposed to enhance distance estimation and measurement [15,16].…”

“…In particular, Lua et al [20] showed that these schemes appear to have poor accuracy under a new relative rank loss metric, which the authors introduced as more meaningful to a number of applications. Lee et al [16] attributed this issue to a large incidence of violations of triangle inequality on the Internet. We include relative rank loss as one of the general evaluation criteria of our approach.…”

“…Lee et al [16] observed that coordinate approaches do not work well for estimating short distances due to large scale of triangle inequality violations. This section considers how the accuracy of our approach depends on the distance between the hosts.…”

Detecting Triangle Inequality Violations in Internet Coordinate Systems by Supervised Learning

“…However, in reality, the triangle inequality is often violated due to the Internet's structure and routing policies. In [7,8,9,10], it has been shown that the violations of triangle inequality (TIVs) in the Internet delay space are not rare and that their impacts on ICS are not negligible. Indeed, a TIV indicates the existence of a shorter path from the source node to the destination node via an intermediate node than a direct path.…”