A Sequential Algorithm for Generating Random Graphs

TL;DR

This paper introduces a nearly-linear time algorithm for efficiently counting and generating random graphs with a specified degree sequence, significantly improving previous methods in speed and range of degrees.

Contribution

It presents a novel nearly-linear time algorithm for uniform graph generation with given degree sequences and provides an independent proof of McKay's enumeration estimate.

Findings

Algorithm runs in O(m d_max) time, faster than previous methods.

Provides FPRAS for counting and generating graphs within the degree range.

Extends uniform generation to higher degrees, improving previous bounds.

Abstract

We present a nearly-linear time algorithm for counting and randomly generating simple graphs with a given degree sequence in a certain range. For degree sequence $(d_i)_{i=1}^n$ with maximum degree $d_{\max}=O(m^{1/4-\tau})$, our algorithm generates almost uniform random graphs with that degree sequence in time $O(m\,d_{\max})$ where $m=\f{1}{2}\sum_id_i$ is the number of edges in the graph and $\tau$ is any positive constant. The fastest known algorithm for uniform generation of these graphs McKay Wormald (1990) has a running time of $O(m^2d_{\max}^2)$. Our method also gives an independent proof of McKay's estimate McKay (1985) for the number of such graphs. We also use sequential importance sampling to derive fully Polynomial-time Randomized Approximation Schemes (FPRAS) for counting and uniformly generating random graphs for the same range of $d_{\max}=O(m^{1/4-\tau})$. Moreover, we show that for $d = O(n^{1/2-\tau})$, our algorithm can generate an asymptotically uniform $d$-regular graph. Our results improve the previous bound of $d = O(n^{1/3-\tau})$ due to Kim and Vu (2004) for regular graphs.