Component sizes in networks with arbitrary degree distributions

TL;DR

This paper provides an exact analytical solution for the distribution of component sizes in random networks with any degree distribution, including applications to epidemic modeling and phase transition analysis.

Contribution

It introduces a comprehensive formula for component size distribution applicable to arbitrary degree distributions in networks.

Findings

Component size distribution follows a power law below the phase transition.

Distribution becomes exponential once a giant component forms.

Results apply to epidemic outbreak sizes and percolation processes.

Abstract

We give an exact solution for the complete distribution of component sizes in random networks with arbitrary degree distributions. The solution tells us the probability that a randomly chosen node belongs to a component of size s, for any s. We apply our results to networks with the three most commonly studied degree distributions -- Poisson, exponential, and power-law -- as well as to the calculation of cluster sizes for bond percolation on networks, which correspond to the sizes of outbreaks of SIR epidemic processes on the same networks. For the particular case of the power-law degree distribution, we show that the component size distribution itself follows a power law everywhere below the phase transition at which a giant component forms, but takes an exponential form when a giant component is present.