The tail does not determine the size of the giant

TL;DR

This paper shows that in the configuration model, the size of the giant component depends more on the distribution of small degrees than on the tail behavior of the degree distribution, challenging common assumptions.

Contribution

It introduces bounds for the giant component size based on small degree distributions and demonstrates their tightness through numerical examples.

Findings

Small degree distribution significantly influences giant component size.

Tail behavior of degree distribution is less crucial for giant component size.

Bounds for component size are effective even with limited degree information.

Abstract

The size of the giant component in the configuration model, measured by the asymptotic fraction of vertices in the component, is given by a well-known expression involving the generating function of the degree distribution. In this note, we argue that the distribution over small degrees is more important for the size of the giant component than the precise distribution over very large degrees. In particular, the tail behavior of the degree distribution does not play the same crucial role for the size of the giant as it does for many other properties of the graph. Upper and lower bounds for the component size are derived for an arbitrary given distribution over small degrees $d\leq L$ and given expected degree, and numerical implementations show that these bounds are close already for small values of $L$. On the other hand, examples illustrate that, for a fixed degree tail, the component size can vary substantially depending on the distribution over small degrees.