Justifying the small-world phenomenon via random recursive trees

TL;DR

This paper introduces a simple, broad, and effective technique using random recursive trees to prove logarithmic upper bounds on diameters of various evolving random graph models, explaining the small-world phenomenon.

Contribution

It presents a new coupling-based method applicable to many models, providing short proofs and tight bounds on graph diameters, including models with preferential attachment.

Findings

Logarithmic upper bounds for diameters of multiple models

Technique applicable to models with preferential attachment

Insights into the small-world phenomenon in real-world graphs

Abstract

We present a new technique for proving logarithmic upper bounds for diameters of evolving random graph models, which is based on defining a coupling between random graphs and variants of random recursive trees. The advantage of the technique is three-fold: it is quite simple and provides short proofs, it is applicable to a broad variety of models including those incorporating preferential attachment, and it provides bounds with small constants. We illustrate this by proving, for the first time, logarithmic upper bounds for the diameters of the following well known models: the forest fire model, the copying model, the PageRank-based selection model, the Aiello-Chung-Lu models, the generalized linear preference model, directed scale-free graphs, the Cooper-Frieze model, and random unordered increasing $k$-trees. Our results shed light on why the small-world phenomenon is observed in so many real-world graphs.