The effect of the initial network configuration on preferential attachment

TL;DR

This paper investigates how the initial network configuration influences the dynamics and degree distribution in preferential attachment models, providing estimates for equilibration time and analyzing the effects of initial density and size.

Contribution

It offers new insights into the sensitivity of preferential attachment to initial conditions, including estimates of equilibration time and the use of weighted Kolmogorov-Smirnov statistic for analysis.

Findings

Dense initial networks prevent equilibration at small sizes.

Equilibration time increases with initial network size and density.

Weighted Kolmogorov-Smirnov statistic effectively detects power-law cutoffs.

Abstract

The classical preferential attachment model is sensitive to the choice of the initial configuration of the network. As the number of initial nodes and their degree grow, so does the time needed for an equilibrium degree distribution to be established. We study this phenomenon, provide estimates of the equilibration time, and characterize the degree distribution cutoff observed at finite times. When the initial network is dense and exceeds a certain small size, there is no equilibration and a suitable statistical test can always discern the produced degree distribution from the equilibrium one. As a by-product, the weighted Kolmogorov-Smirnov statistic is demonstrated to be more suitable for statistical analysis of power-law distributions with cutoff when the data is ample.