2007/02/28 by Oskar J. Sandberg, Oskar Sandberg · 4 citations
Computer Science · Mathematics · Physics and Astronomy · #Algorithms and Data Compression #Caching and Content Delivery #Complex Network Analysis Techniques #cs.DS #math.PR #msc:60C05 #msc:68R10 #msc:68W20
paper · pdf · doi:10.1214/07-aap499
published as Annals of Applied Probability 2008, Vol. 18, No. 5, 1771-1793 · Published in at http://dx.doi.org/10.1214/07-AAP499 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
arxiv created 2008/11/12 · arxiv updated 2009/12/01
Small-world graphs, which combine randomized and structured elements, are seen as prevalent in nature. Jon Kleinberg showed that in some graphs of this type it is possible to route, or navigate, between vertices in few steps even with very little knowledge of the graph itself. In an attempt to understand how such graphs arise we introduce a different criterion for graphs to be navigable in this sense, relating the neighbor selection of a vertex to the hitting probability of routed walks. In several models starting from both discrete and continuous settings, this can be shown to lead to graphs with the desired properties. It also leads directly to an evolutionary model for the creation of similar graphs by the stepwise rewiring of the edges, and we conjecture, supported by simulations, that these too are navigable.