A Study on the Amount of Random Graph Groupies

TL;DR

This paper investigates the proportion of 'groupies' in random graphs and bipartite graphs, showing that in large Erdős–Rényi graphs, this proportion converges to approximately 84.13%, with additional analysis on bipartite cases.

Contribution

It provides a probabilistic analysis of the asymptotic behavior of groupies in random and bipartite graphs, extending the understanding of local vertex properties in these models.

Findings

Proportion of groupies in G(n,p) converges to Φ(1)≈0.8413.

Asymptotic behavior characterized for complete bipartite graphs.

Results connect local degree properties with global graph structure.

Abstract

In 1980, Ajtai, Komlos and Szemer{\'e}di defined "groupie": Let $G=(V,E)$ be a simple graph, $|V|=n$, $|E|=e$. For a vertex $v\in V$, let $r(v)$ denote the sum of the degrees of the vertices adjacent to $v$. We say $v\in V$ is a {\it groupie}, if $\frac{r(v)}{\deg(v)}\geq\frac{e}{n}.$ In this paper, we prove that in random graph $B(n,p)$, $0<p<1$, the proportion of groupies converges in probability towards $\Phi(1)\approx0.8413$ as $n$ approaches infinity, where $\Phi(x)$ is the distribution function of standard normal distribution N(0,1). We also discuss the asymptotic behavior of the proportion of groupies in complete bipartite graph $B(n_1,n_2,p)$.