# Fast uniform generation of random graphs with given degree sequences

## TL;DR

This paper introduces a highly efficient algorithm for uniformly generating random graphs with specified degree sequences, significantly improving upon previous methods in terms of speed and applicability to various degree distributions.

## Contribution

The authors present a novel algorithm that achieves expected linear time complexity for generating graphs with given degree sequences under certain conditions, advancing the state of the art.

## Key findings

- Expected runtime is $O(m)$ for graphs with $ ext{max degree}^4=O(m)$.
- Algorithm outperforms previous $O(m^2 ext{max degree}^2)$ methods.
- Effective for power-law and $d$-regular degree sequences, reducing computational complexity.

## Abstract

In this paper we provide an algorithm that generates a graph with given degree sequence uniformly at random. Provided that $\Delta^4=O(m)$, where $\Delta$ is the maximal degree and $m$ is the number of edges,the algorithm runs in expected time $O(m)$. Our algorithm significantly improves the previously most efficient uniform sampler, which runs in expected time $O(m^2\Delta^2)$ for the same family of degree sequences. Our method uses a novel ingredient which progressively relaxes restrictions on an object being generated uniformly at random, and we use this to give fast algorithms for uniform sampling of graphs with other degree sequences as well. Using the same method, we also obtain algorithms with expected run time which is (i) linear for power-law degree sequences in cases where the previous best was $O(n^{4.081})$, and (ii) $O(nd+d^4)$ for $d$-regular graphs when $d=o(\sqrt n)$, where the previous best was $O(nd^3)$.