An elementary proof of the total progeny size of a birth-death process, with application to network component sizes

TL;DR

This paper presents an elementary combinatorial proof for the size distribution of birth-death trees, offering a more intuitive understanding of component sizes in infinite networks, and demonstrates its robustness on heavy-tailed networks.

Contribution

It provides a simpler, more intuitive proof for the component size distribution in configuration model networks, improving understanding and applicability.

Findings

Elementary proof enhances understanding of birth-death tree sizes

Derived component size distribution matches empirical data

Method remains effective on heavy-tailed networks

Abstract

We revisit the size distribution of finite components in infinite Configuration Model networks. We provide an elementary combinatorial proof about the sizes of birth-death trees which is more intuitive than previous proofs. We use this to rederive the component size distribution for Configuration Model networks. Our derivation provides a more intuitive interpretation of the formula as contrasted with the previous derivation based on contour integrations. We demonstrate that the formula performs well, even on networks with heavy tails which violate assumptions of the derivation. We explain why the result should remain robust for these networks.