Simple rules govern finite-size effects in scale-free networks

TL;DR

This paper explains how finite-size effects in scale-free networks are influenced by the initial network's degree distribution, using mean-field and stochastic approaches, with simulations validating the predictions.

Contribution

It provides a general explanation of finite-size effects in scale-free networks considering the starting network, combining mean-field and stochastic methods.

Findings

Mean-field approximation's accuracy depends on dispersion in the degree distribution.

Stochastic approach yields highly accurate predictions compared to simulations.

Starting core significantly influences the degree distribution in finite networks.

Abstract

We give an intuitive though general explanation of the finite-size effect in scale-free networks in terms of the degree distribution of the starting network. This result clarifies the relevance of the starting network in the final degree distribution. We use two different approaches: the deterministic mean-field approximation used by Barab\'asi and Albert (but taking into account the nodes of the starting network), and the probability distribution of the degree of each node, which considers the stochastic process. Numerical simulations show that the accuracy of the predictions of the mean-field approximation depend on the contribution of the dispersion in the final distribution. The results in terms of the probability distribution of the degree of each node are very accurate when compared to numerical simulations. The analysis of the standard deviation of the degree distribution allows us to assess the influence of the starting core when fitting the model to real data.