Typical distances in ultrasmall random networks

TL;DR

This paper analyzes the typical distances in ultrasmall random networks generated by preferential attachment models with power-law degree distributions, revealing a doubled logarithmic scale compared to configuration models.

Contribution

It provides an asymptotic formula for typical distances in preferential attachment networks with degree exponent between 2 and 3, highlighting structural differences from configuration models.

Findings

Typical distance scales as (4+o(1)) * log log N / -log(τ-2)

Distance is twice that of configuration models with same exponent

Shows structural differences in shortest paths in preferential attachment graphs

Abstract

We show that in preferential attachment models with power-law exponent $\tau\in(2,3)$ the distance between randomly chosen vertices in the giant component is asymptotically equal to $(4+o(1))\, \frac{\log\log N}{-\log (\tau-2)}$, where $N$ denotes the number of nodes. This is twice the value obtained for several types of configuration models with the same power-law exponent. The extra factor reveals the different structure of typical shortest paths in preferential attachment graphs.