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A Sequential Algorithm for Generating Random Graphs

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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.

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−τ).

Moreover, we show that for d=O(n 1/2−τ), our algorithm can generate an asymptotically uniform d-regular graph. Our results improve the previous bound of d=O(n 1/3−τ) due to Kim and Vu (Adv. Math. 188:444–469, 2004) for regular graphs.

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Authors and Affiliations

  1. Department of Electrical Engineering, Stanford University, Stanford, CA, 94305-9510, USA

    Mohsen Bayati

  2. Department of Mathematics, Yonsei University, Yonsei, South Korea

    Jeong Han Kim

  3. Departments of Management Science and Engineering, Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA, 94305, USA

    Amin Saberi

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  1. Mohsen Bayati
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  2. Jeong Han Kim
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  3. Amin Saberi
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Correspondence to Jeong Han Kim.

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Bayati, M., Kim, J.H. & Saberi, A. A Sequential Algorithm for Generating Random Graphs. Algorithmica 58, 860–910 (2010). https://doi.org/10.1007/s00453-009-9340-1

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