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Navigability is a Robust Property

  • Dimitris Achlioptas
  • Paris SiminelakisEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9479)

Abstract

The Small World phenomenon has inspired researchers across a number of fields. A breakthrough in its understanding was made by Kleinberg who introduced Rank Based Augmentation (RBA): add to each vertex independently an arc to a random destination, selected from a carefully crafted probability distribution. Kleinberg proved that RBA makes many networks navigable, i.e., it allows greedy routing to successfully deliver messages between any two vertices in a polylogarithmic number of steps. Our goal in this work is to prove that navigability is an inherent, robust property of many random networks, requiring no augmentation, coordination, or even independence assumptions. Our framework assigns a cost to each edge and considers the uniform measure over all graphs on n vertices that satisfy a total budget constraint. We show that when the cost function is sufficiently correlated with the underlying geometry of the vertices and for a wide range of budgets, the overwhelming majority of all feasible graphs with the given budget are navigable. We provide a new set of geometric conditions that generalize Kleinberg’s set systems as well as a unified analysis of navigability.

Keywords

Cost Function Random Graph Product Measure Distance Scale Uniform Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of CaliforniaSanta CruzUSA
  2. 2.Department of Electrical EngineeringStanford UniversityStanfordUSA

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