On the growth of components with non fixed excesses

TL;DR

This paper analyzes the growth dynamics of connected graph components with excess edges, showing asymptotic behaviors of their creation and size in large random graphs with specific growth conditions.

Contribution

It provides new asymptotic results on the expected number of component creations and their sizes in large random graphs with non-fixed excess edges.

Findings

Expected number of (l+1)-components tends to 1 as n and l grow under certain conditions.

Expected size of vertices in l-components scales as (12l)^{1/3} n^{2/3}.

Results hold for l=o(n^{1/4}) in large random graphs.

Abstract

Denote by an $l$-component a connected graph with $l$ edges more than vertices. We prove that the expected number of creations of $(l+1)$-component, by means of adding a new edge to an $l$-component in a randomly growing graph with $n$ vertices, tends to 1 as $l,n$ tends to $\infty$ but with $l = o(n^{1/4})$. We also show, under the same conditions on $l$ and $n$, that the expected number of vertices that ever belong to an $l$-component is $\sim (12l)^{1/3} n^{2/3}$.