TY - JOUR
T1 - On suitability of euclidean embedding for host-based network coordinate systems
AU - Lee, Sanghwan
AU - Zhang, Zhi Li
AU - Sahu, Sambit
AU - Saha, Debanjan
PY - 2010/2/1
Y1 - 2010/2/1
N2 - 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 two-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 seven-dimensional Euclidean embedding.
AB - 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 two-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 seven-dimensional Euclidean embedding.
KW - Euclidean embedding
KW - Suitability
KW - Triangle inequality
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U2 - 10.1109/TNET.2009.2023322
DO - 10.1109/TNET.2009.2023322
M3 - Article
AN - SCOPUS:77249105918
VL - 18
SP - 27
EP - 40
JO - IEEE/ACM Transactions on Networking
JF - IEEE/ACM Transactions on Networking
SN - 1063-6692
IS - 1
M1 - 5235089
ER -