Abstract This article introduces new efficient algorithms for two problems: sampling conditional on vertex degrees in unweighted graphs, and conditional on vertex strengths in weighted graphs. The resulting conditional distributions provide the basis for exact tests on social networks and two-way contingency tables. The algorithms are able to sample conditional on the presence or absence of an arbitrary set of edges. Existing samplers based on MCMC or sequential importance sampling are generally not scalable; their efficiency can degrade in large graphs with complex patterns of known edges. MCMC methods usually require explicit computation of a Markov basis to navigate the state space; this is computationally intensive even for small graphs. Our samplers do not require a Markov basis, and are efficient both in sparse and dense settings. The key idea is to carefully select a Markov kernel on the basis of the current state of the chain. We demonstrate the utility of our methods on a real network and contingency table. Supplementary materials for this article are available online.

中文翻译：

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图的条件采样的状态相关内核选择

摘要 本文针对两个问题介绍了新的高效算法：未加权图中以顶点度为条件的采样，以及加权图中以顶点强度为条件的采样。由此产生的条件分布为社交网络和双向列联表的精确测试提供了基础。该算法能够以任意一组边的存在或不存在为条件进行采样。现有的基于 MCMC 或顺序重要性采样的采样器通常不可扩展；在具有复杂已知边模式的大型图中，它们的效率会降低。MCMC 方法通常需要显式计算马尔可夫基来导航状态空间；即使对于小图，这也是计算密集型的。我们的采样器不需要马尔可夫基，并且在稀疏和密集设置中都很有效。关键思想是根据链的当前状态仔细选择马尔可夫核。我们在真实的网络和列联表上展示了我们的方法的实用性。本文的补充材料可在线获取。