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

TL;DR

This paper presents a detailed description and implementation of Inc-Powerlaw, a novel algorithm for uniformly sampling graphs with power-law degree sequences, offering rigorous guarantees and high efficiency especially for small average degrees.

Contribution

It provides the first complete description and open-source implementation of Inc-Powerlaw, a new algorithm with rigorous uniformity guarantees for power-law degree sequences.

Findings

Inc-Powerlaw is highly efficient for small average degrees.

The implementation can generate graphs with one million nodes in less than a second.

Parallelism helps reduce running time for larger average degrees.

Abstract

We consider the following common network analysis problem: given a degree sequence $\mathbf{d} = (d_1, \dots, d_n) \in \mathbb N^n$ 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 $\gamma \gtrapprox 2.88$ 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.