Analytic solutions for links and triangles distributions in finite Barab\'asi-Albert networks

TL;DR

This paper develops and solves master equations for degree, link, and triangle distributions in finite Barabási-Albert networks, providing analytical solutions that match simulations even for small networks, enhancing understanding of finite size effects.

Contribution

It introduces a novel analytical framework for finite Barabási-Albert networks, enabling precise predictions of network distributions.

Findings

Analytical solutions match simulations for networks as small as 100 nodes.

Master equations accurately describe the evolution of degrees, links, and triangles.

The method is applicable to other network classes, aiding the study of natural networks.

Abstract

Barab\'asi-Albert model describes many different natural networks, often yielding sensible explanations to the subjacent dynamics. However, finite size effects may prevent from discerning among different underlying physical mechanisms and from determining whether a particular finite system is driven by Barab\'asi-Albert dynamics. Here we propose master equations for the evolution of the degrees, links and triangles distributions, solve them both analytically and by numerical iteration, and compare with numerical simulations. The analytic solutions for all these distributions predict the network evolution for systems as small as 100 nodes. The analytic method we developed is applicable for other classes of networks, representing a powerful tool to investigate the evolution of natural networks.