A Sequential Algorithm for Generating Random Graphs

Download Book (11,611 KB) As a courtesy to our readers the eBook is provided DRM-free. However, please note that Springer uses effective methods and state-of-the art technology to detect, stop, and prosecute illegal sharing to safeguard our authors’ interests.
Download Chapter (511 KB)

Abstract

We present the fastest FPRAS 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 graph with that degree sequence in time O(m d max ) where is the number of edges in the graph and τ is any positive constant. The fastest known FPRAS for this problem [22] has running time of O(m 3 n 2). Our method also gives an independent proof of McKay’s estimate [33] for the number of such graphs.

Our approach is based on sequential importance sampling (SIS) technique that has been recently successful for counting graphs [15,11,10]. Unfortunately validity of the SIS method is only known through simulations and our work together with [10] are the first results that analyze the performance of this method.

Moreover, we show that for d = O(n 1/2 − τ ), our algorithm can generate an asymptotically uniform d-regular graph. Our results are improving the previous bound of d = O(n 1/3 − τ ) due to Kim and Vu [30] for regular graphs.