Engineering Uniform Sampling of Graphs with a Prescribed Power-law Degree Sequence

Abstract

We consider the following common network analysis problem: given a degree sequence d = (d1,…, dn) ∈ ℕn return a uniform sample from the ensemble of all simple graphs with matching degrees. In practice, the problem is typically solved using Markov Chain Monte Carlo approaches, such as Edge-Switching or Curveball, even if no practical useful rigorous bounds are known on their mixing times. In contrast, Arman et al. sketch INC-PoWERLAW, a novel and much more involved algorithm capable of generating graphs for power-law bounded degree sequences with γ ⪆ 2.88 in expected linear time. For the first time, we give a complete description of the algorithm and add novel switchings. To the best of our knowledge, our open-source implementation of INC-POWERLAW is the first practical generator with rigorous uniformity guarantees for the aforementioned degree sequences. In an empirical investigation, we find that for small average-degrees INC-POWERLAW is very efficient and generates graphs with one million nodes in less than a second. For larger average-degrees, parallelism can partially mitigate the increased running-time.