What is The Probability That A Random Graph With A Given Degree Sequence is Connected?
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
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2026
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| _version_ | 1866917444154556416 |
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| author | Addario-Berry, Louigi Reed, Bruce Yuan, Dao Chen |
| author_facet | Addario-Berry, Louigi Reed, Bruce Yuan, Dao Chen |
| contents | 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 $O(\frac{n^4}{m^6})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_25725 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | What is The Probability That A Random Graph With A Given Degree Sequence is Connected? Addario-Berry, Louigi Reed, Bruce Yuan, Dao Chen Probability Combinatorics 60C05, 05C80 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 $O(\frac{n^4}{m^6})$. |
| title | What is The Probability That A Random Graph With A Given Degree Sequence is Connected? |
| topic | Probability Combinatorics 60C05, 05C80 |
| url | https://arxiv.org/abs/2604.25725 |