# General expression for the component size distribution in infinite configuration networks

## TL;DR

This paper derives a universal equation for the component size distribution in infinite configuration networks, linking it to degree distribution moments and tail properties, enabling efficient computation and analysis.

## Contribution

It introduces a simple, universal equation for component size distribution in infinite configuration networks, applicable to arbitrary degree distributions and suitable for fast numerical analysis.

## Key findings

- Component size distribution asymptote depends on degree distribution moments.
- Heavy-tailed degree distributions can produce multiple asymptotic modes.
- The equation allows stable numerical computation up to machine precision.

## Abstract

In the infinite configuration network the links between nodes are assigned randomly with the only restriction that the degree distribution has to match a predefined function. This work presents a simple equation that gives for an arbitrary degree distribution the corresponding size distribution of connected components. This equation is suitable for fast and stable numerical computations up to the machine precision. The analytical analysis reveals that the asymptote of the component size distribution is completely defined by only a few parameters of the degree distribution: the first three moments, scale and exponent (if applicable). When the degree distribution features a heavy tail, multiple asymptotic modes are observed in the component size distribution that, in turn, may or may not feature a heavy tail.