A Sequential Algorithm for Generating Random Graphs

Mohsen Bayati, Jeong Han Kim, Amin Saberi

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 ) ni=1 with maximum degree d max =O(m 1/4−τ), our algorithm generates almost uniform random graphs with that degree sequence in time O(md max ) where \(m=\frac{1}{2}\sum_{i}d_{i}\) is the number of edges in the graph and τ is any positive constant. The fastest known algorithm for uniform generation of these graphs (McKay and Wormald in J. Algorithms 11(1):52–67, 1990) has a running time of O(m 2 d 2max ). Our method also gives an independent proof of McKay’s estimate (McKay in Ars Combinatoria A 19:15–25, 1985) for the number of such graphs.