What is The Probability That A Random Graph With A Given Degree Sequence is Connected?

TL;DR

This paper investigates the probability that a random graph with a specified degree sequence is connected, providing bounds based on the count of low-degree vertices and developing tools for such probabilistic analysis.

Contribution

It introduces a method to upper-bound the connectivity probability of graphs with given degree sequences, especially considering vertices of small degrees.

Findings

If no vertices of degree zero, the probability of disconnection depends on degree 1 and 2 vertices.

High probability of connectivity if vertices of degree 1 are o(√m) and degree 2 are o(m).

Probability of disconnection is very low (O(n^4/m^6)) if no vertices of degree 1 or 2.

Abstract

An $n$-tuple $D=(d(1),\dots,d(n))$ is a \emph{feasible degree sequence} if there is a graph on $\{1,\dots,n\}$ such that $i$ has degree $d(i)$. Any such graph will have $m=\sum_{i=1}^n d(i)/2$ edges. Letting $G(D)$ be a graph chosen uniformly from those with the given degree sequence, we upper-bound the probability that $G(D)$ is disconnected based on the number of vertices of degree $d$ for small $d$, and develop a powerful tool for proving such bounds. If there are any vertices of degree zero the probability $G$ is disconnected is $1$, so we assume there are no such vertices. Our results then imply that if there are $o(\sqrt{m})$ vertices of degree $1$ and $o(m)$ vertices of degree 2 then with high probability $G$ is connected, while if there are no vertices of degree 1 or 2 then the probability $G$ is disconnected is $O(\frac{n^4}{m^6})$.