Navigating ultrasmall worlds in ultrashort time

TL;DR

This paper demonstrates that ultrasmall scale-free networks can be navigated efficiently using local information, achieving near-optimal path lengths despite the lack of global topology knowledge, with significant implications for scalable communication systems.

Contribution

It shows that greedy routing in embedded scale-free networks can find asymptotically shortest paths using only local information, matching the lnlnN scaling of global shortest paths.

Findings

Greedy routing achieves lnlnN path length scaling.

Local information suffices for near-optimal navigation.

Implications for scalable Internet communication systems.

Abstract

Random scale-free networks are ultrasmall worlds. The average length of the shortest paths in networks of size N scales as lnlnN. Here we show that these ultrasmall worlds can be navigated in ultrashort time. Greedy routing on scale-free networks embedded in metric spaces finds paths with the average length scaling also as lnlnN. Greedy routing uses only local information to navigate a network. Nevertheless, it finds asymptotically the shortest paths, a direct computation of which requires global topology knowledge. Our findings imply that the peculiar structure of complex networks ensures that the lack of global topological awareness has asymptotically no impact on the length of communication paths. These results have important consequences for communication systems such as the Internet, where maintaining knowledge of current topology is a major scalability bottleneck.