A Sequential Algorithm for Generating Random Graphs

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 $(d_i)<em>{i=1}<sup>n$ with maximum degree $d</em>{\max}=O(m<sup>{1/4-\tau})$, 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 $O(m<sup>2d_{\max}<sup>2)$. 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 $d_{\max}=O(m<sup>{1/4-\tau})$. Moreover, we show that for $d = O(n<sup>{1/2-\tau})$, our algorithm can generate an asymptotically uniform $d$-regular graph. Our results improve the previous bound of $d = O(n<sup>{1/3-\tau})$ due to Kim and Vu (2004) for regular graphs.