Uniform generation of random graphs with power-law degree sequences

TL;DR

This paper introduces a linear-time algorithm for approximately uniform generation of simple graphs with power-law degree sequences having an exponent at least 2.8811, addressing a challenging case where the exponent is below 3, and also provides an exact sampling method.

Contribution

It presents the first provably practical algorithm for uniform graph sampling with power-law exponents below 3, extending the applicability of such sampling methods.

Findings

Linear-time approximate uniform sampler for exponents ≥ 2.8811

Exact uniform sampler with polynomial runtime

Addresses a previously difficult regime for power-law exponents

Abstract

We give a linear-time algorithm that approximately uniformly generates a random simple graph with a power-law degree sequence whose exponent is at least 2.8811. While sampling graphs with power-law degree sequence of exponent at least 3 is fairly easy, and many samplers work efficiently in this case, the problem becomes dramatically more difficult when the exponent drops below 3; ours is the first provably practicable sampler for this case. We also show that with an appropriate rejection scheme, our algorithm can be tuned into an exact uniform sampler. The running time of the exact sampler is O(n^{2.107}) with high probability, and O(n^{4.081}) in expectation.