Diameters in preferential attachment models

TL;DR

This paper analyzes the diameter of preferential attachment models, showing it scales as log t for au>3 and as log log t for au in (2,3), confirming physicists' predictions about small-world properties.

Contribution

It provides rigorous bounds on the diameter of affine PA-models, establishing the order of typical distances for different degree exponent regimes.

Findings

Diameter is bounded by a constant times log t for au>3.

Diameter and typical distance are of order log log t for au in (2,3).

Results align with predictions for scale-free networks' small-world behavior.

Abstract

In this paper, we investigate the diameter in preferential attachment (PA-) models, thus quantifying the statement that these models are small worlds. The models studied here are such that edges are attached to older vertices proportional to the degree plus a constant, i.e., we consider affine PA-models. There is a substantial amount of literature proving that, quite generally, PA-graphs possess power-law degree sequences with a power-law exponent \tau>2. We prove that the diameter of the PA-model is bounded above by a constant times \log{t}, where t is the size of the graph. When the power-law exponent \tau exceeds 3, then we prove that \log{t} is the right order, by proving a lower bound of this order, both for the diameter as well as for the typical distance. This shows that, for \tau>3, distances are of the order \log{t}. For \tau\in (2,3), we improve the upper bound to a constant times \log\log{t}, and prove a lower bound of the same order for the diameter. Unfortunately, this proof does not extend to typical distances. These results do show that the diameter is of order \log\log{t}. These bounds partially prove predictions by physicists that the typical distance in PA-graphs are similar to the ones in other scale-free random graphs, such as the configuration model and various inhomogeneous random graph models, where typical distances have been shown to be of order \log\log{t} when \tau\in (2,3), and of order \log{t} when \tau>3.