A critique of the Mean Field Approximation in preferential attachment networks

TL;DR

This paper critically examines the Mean Field Approximation in preferential attachment networks, revealing its flaws, unresolvable contradictions, and lack of rigorous derivation, questioning its suitability for educational purposes.

Contribution

It provides a detailed critique showing that MFA's success is accidental and highlights the need for more rigorous derivations in network analysis.

Findings

MFA leads to unresolvable contradictions.

MFA is not explicitly derived from a proper model.

The success of MFA is largely accidental.

Abstract

The Mean Field Approximation (MFA), or continuum method, is often used in courses on Networks to derive the degree distribution of preferential attachment networks. This method is simple and the outcome is close to the correct answer. However, this paper shows that the method is flawed in several aspects, leading to unresolvable contradictions. More importantly, the MFA is not explicitly derived from a mathematical model. An analysis of the implied model shows that it makes an approximation which is far from the truth and another one which can not be motivated in general. The success of the MFA for preferential attachment networks is therefore accidental and the method is not suitable for teaching undergraduates.