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Polylogarithmic network navigability using compact metrics with small stretch

Published: 14 June 2008 Publication History

Abstract

Graph augmentation theory is a general framework for analyzing navigability in social networks. It is known that, for large classes of graphs, there exist augmentations of these graphs such that greedy routing according to the shortest path metric performs in polylogarithmic expected number of steps. However, it is also known that there are classes of graphs for which no augmentations can enable greedy routing according to the shortest path metric to perform better than Ω(n1/√log n) expected number of steps. In fact, the best known universal bound on the greedy diameter of arbitrary graph is essentially n1/3. That is, for any graph, there is an augmentation such that greedy routing according to the shortest path metric performs in Õ(n1/3) expected number of steps. Hence, greedy routing according to the shortest path metric has at least two drawbacks. First, it is in general space-consuming to encode locally the shortest path distances to all the other nodes, and, second, greedy routing according to the shortest path metric performs poorly in some graphs.
We prove that, using semimetrics of small stretch results in a huge positive impact, in both encoding space and efficiency of greedy routing. More precisely, we show that, for any connected n-node graph G and any integer k ≥ 1, there exist an augmentation φ of G and a semimetric μ on G with stretch 2/k-1 such that greedy routing according to μ performs in O(k2 n2/klog2n) expected number of steps. As a corollary, we get that for any connected n-node graph G, there exist an augmentation φ of G and a semimetric μ on G with stretch O(log n) such that greedy routing according to μ performs in polylogarithmic expected number of steps. This latter semimetric can be encoded locally at every node using only a polylogarithmic number of bits.

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Cited By

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  • (2010)An experimental study of greedy routing algorithms2010 International Conference on High Performance Computing & Simulation10.1109/HPCS.2010.5547143(150-156)Online publication date: Jun-2010
  • (2009)Graph embedding through random walk for shortest paths problemsProceedings of the 5th international conference on Stochastic algorithms: foundations and applications10.5555/1814087.1814101(127-140)Online publication date: 26-Oct-2009
  • (2009)Universal augmentation schemes for network navigabilityTheoretical Computer Science10.1016/j.tcs.2008.12.061410:21-23(1970-1981)Online publication date: 1-May-2009
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cover image ACM Conferences
SPAA '08: Proceedings of the twentieth annual symposium on Parallelism in algorithms and architectures
June 2008
380 pages
ISBN:9781595939739
DOI:10.1145/1378533
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 14 June 2008

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Author Tags

  1. network navigability
  2. small world phenomenon
  3. social networks

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SPAA08

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Overall Acceptance Rate 447 of 1,461 submissions, 31%

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Cited By

View all
  • (2010)An experimental study of greedy routing algorithms2010 International Conference on High Performance Computing & Simulation10.1109/HPCS.2010.5547143(150-156)Online publication date: Jun-2010
  • (2009)Graph embedding through random walk for shortest paths problemsProceedings of the 5th international conference on Stochastic algorithms: foundations and applications10.5555/1814087.1814101(127-140)Online publication date: 26-Oct-2009
  • (2009)Universal augmentation schemes for network navigabilityTheoretical Computer Science10.1016/j.tcs.2008.12.061410:21-23(1970-1981)Online publication date: 1-May-2009
  • (2009)Graph Embedding through Random Walk for Shortest Paths ProblemsStochastic Algorithms: Foundations and Applications10.1007/978-3-642-04944-6_11(127-140)Online publication date: 2009
  • (2008)Graph Augmentation via Metric EmbeddingProceedings of the 12th International Conference on Principles of Distributed Systems10.1007/978-3-540-92221-6_15(217-225)Online publication date: 15-Dec-2008

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