Greedy Routing and the Algorithmic Small-World Phenomenom

TL;DR

This paper introduces a flexible random graph model that better explains the algorithmic small-world phenomenon, demonstrating that decentralized greedy routing can efficiently find near-optimal paths with high probability, addressing longstanding theoretical questions.

Contribution

It provides a new, more realistic model for analyzing greedy routing, proving its effectiveness and success probability in finding short paths in complex networks.

Findings

Greedy routing succeeds with constant probability in the new model.

Almost shortest paths of length loglog n are found with high probability.

Local patching methods can guarantee success with asymptotically optimal steps.

Abstract

The algorithmic small-world phenomenon, empirically established by Milgram's letter forwarding experiments from the 60s, was theoretically explained by Kleinberg in 2000. However, from today's perspective his model has several severe shortcomings that limit the applicability to real-world networks. In order to give a more convincing explanation of the algorithmic small-world phenomenon, we study decentralized greedy routing in a more flexible random graph model (geometric inhomogeneous random graphs) which overcomes all previous shortcomings. Apart from exhibiting good properties in theory, it has also been extensively experimentally validated that this model reasonably captures real-world networks. In this model, the greedy routing protocol is purely distributed as each vertex only needs to know information about its direct neighbors. We prove that it succeeds with constant probability, and in case of success almost surely finds an almost shortest path of length {\theta}(loglog n), where our bound is tight including the leading constant. Moreover, we study natural local patching methods which augment greedy routing by backtracking and which do not require any global knowledge. We show that such methods can ensure success probability 1 in an asymptotically tight number of steps. These results also address the question of Krioukov et al. whether there are efficient local routing protocols for the internet graph. There were promising experimental studies, but the question remained unsolved theoretically. Our results give for the first time a rigorous and analytical affirmative answer.