The uniform sampling of simple graphs matching a prescribed degree sequence is an important tool in network science, e.g. to construct graph generators or null-models. Here, the Edge Switching Markov Chain (ES-MC) is a common choice. Given an arbitrary simple graph with the required degree sequence, ES-MC carries out a large number of small changes, called edge switches, to eventually obtain a uniform sample. In practice, reasonably short runs efficiently yield approximate uniform samples. In this work, we study the problem of executing edge switches in parallel. We discuss parallelizations of ES-MC, but find that this approach suffers from complex dependencies between edge switches. For this reason, we propose the Global Edge Switching Markov Chain (G-ES-MC), an ES-MC variant with simpler dependencies. We show that G-ES-MC converges to the uniform distribution and design shared-memory parallel algorithms for ES-MC and G-ES-MC. In an empirical evaluation, we provide evidence that G-ES-MC requires not more switches than ES-MC (and often fewer), and demonstrate the efficiency and scalability of our parallel G-ES-MC implementation.

翻译：对具有指定度数序列的简单图进行均匀采样是网络科学中的重要工具，例如用于构建图生成器或零模型。其中，边交换马尔可夫链（ES-MC）是一种常见选择。给定一个具有所需度数序列的任意简单图，ES-MC会执行大量称为边交换的微小修改，最终获得均匀样本。在实践中，适中的运行长度可高效地获得近似均匀样本。本文研究了并行执行边交换的问题。我们讨论了ES-MC的并行化方案，但发现该方法存在边交换之间的复杂依赖关系。为此，我们提出了全局边交换马尔可夫链（G-ES-MC），一种具有更简单依赖关系的ES-MC变体。我们证明了G-ES-MC收敛于均匀分布，并为ES-MC和G-ES-MC设计了共享内存并行算法。在实证评估中，我们证明了G-ES-MC所需的交换次数不多于ES-MC（通常更少），并展示了并行G-ES-MC实现的效率和可扩展性。