Type of Publication: Journal Articles
Authors: Zvi Lotker,
Title: Universal augmentation schemes for network navigability
Name of the Journal: Theoretical Computer Science
Year: 2009
Volume: 410
Issue: 21-23
Pages: 1961-2300

Augmented graphs were introduced for the purpose of analyzing the “six degrees of separation between individuals” observed experimentally by the sociologist Standley Milgram in the 60’s. We define an augmented graph as a pair (G,M) where G is an n-node graph with nodes labeled in {1,…,n}, and M is an n×n stochastic matrix. Every node uset membership, variantV(G) is given an extra link, called a long range link, pointing to some node v, called the long range contact of u. The head v of this link is chosen at random by View the MathML source. In augmented graphs, greedy routing is the oblivious routing process in which every intermediate node chooses from among all its neighbors (including its long range contact) the one that is closest to the target according to the distance measured in the underlying graph G, and forwards to it. The best augmentation scheme known so far ensures that, for any n-node graph G, greedy routing performs in View the MathML source expected number of steps.

Our main result is the design of an augmentation scheme that overcomes the View the MathML source barrier. Precisely, we prove that for any n-node graph G whose nodes are arbitrarily labeled in {1,…,n}, there exists a stochastic matrix M such that greedy routing in (G,M) performs in View the MathML source, where the View the MathML source notation ignores the polylogarithmic factors.

We prove additional results when the stochastic matrix M is universal to all graphs. In particular, we prove that the View the MathML source barrier can still be overcame for large graph classes even if the matrix M is universal. This however requires an appropriate labeling of the nodes. If the node labeling is arbitrary, then we prove that the View the MathML source barrier cannot be overcome with universal matrices.

Keywords: Small world phenomenon; Informative labeling schemes; Routing
Last Updated: 8/29/2009 4:54:04 PM
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