A lower bound for network navigability

Pierre Fraigniaud, Emmanuelle Lebhar, Zvi Lotker

Research output: Contribution to journalArticlepeer-review

8 Scopus citations

Abstract

In his seminal work, Kleinberg showed how to augment meshes using random edges, so that they become navigable; that is, greedy routing computes paths of polylogarithmic expected length between any pairs of nodes. This yields the crucial question of determining whether such an augmentation is possible for all graphs. In this paper, we answer this question negatively by exhibiting an infinite family of graphs that cannot be augmented to become navigable whatever the distribution of random edges is. Precisely, it was known that graphs of doubling dimension at most O(log log n) are navigable. We show that for doubling dimension » log log n, an infinite family of graphs cannot be augmented to become navigable. Finally, we present a positive navigability result by studying the special case of square meshes of arbitrary dimension that we prove to always be augmentable to become navigable. This latter result complements Kleinberg's original result and shows that adding extra links can sometimes break the navigability.

Original languageEnglish
Pages (from-to)72-81
Number of pages10
JournalSIAM Journal on Discrete Mathematics
Volume24
Issue number1
DOIs
StatePublished - 24 May 2010

Keywords

  • Doubling dimension
  • Greedy routing
  • Small world

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