Maximizing the size of the giant

TL;DR

This paper investigates how to maximize the size of the giant component in two types of random graphs by choosing optimal degree or weight distributions, revealing specific distributions that achieve this maximum.

Contribution

It identifies the optimal distributions of weights and degrees that maximize the giant component size in Poissonian and thinned configuration models.

Findings

Optimal weight distribution in Poissonian graphs is concentrated at 0 and one other point.

Optimal degree distribution in thinned configuration models is concentrated at 0 and two consecutive integers.

Maximizing the giant component depends on selecting specific discrete distributions.

Abstract

We consider two classes of random graphs: $(a)$ Poissonian random graphs in which the $n$ vertices in the graph have i.i.d.\ weights distributed as $X$, where $\mathbb{E}(X) = \mu$. Edges are added according to a product measure and the probability that a vertex of weight $x$ shares and edge with a vertex of weight $y$ is given by $1-e^{-xy/(\mu n)}$. $(b)$ A thinned configuration model in which we create a ground-graph in which the $n$ vertices have i.i.d.\ ground-degrees, distributed as $D$, with $\mathbb{E}(D) = \mu$. The graph of interest is obtained by deleting edges independently with probability $1-p$. In both models the fraction of vertices in the largest connected component converges in probability to a constant $1-q$, where $q$ depends on $X$ or $D$ and $p$. We investigate for which distributions $X$ and $D$ with given $\mu$ and $p$, $1-q$ is maximized. We show that in the class of Poissonian random graphs, $X$ should have all its mass at 0 and one other real, which can be explicitly determined. For the thinned configuration model $D$ should have all its mass at 0 and two subsequent positive integers.