度序列给定的有向图均匀生成的序贯短截线匹配
Sequential stub matching for uniform generation of directed graphs with a given degree sequence
摘要：
具有给定度序列的单图的均匀采样是一个已知的问题，其平均度的平方具有指数复杂性。对于无向图，随机近似算法已经被证明可以实现几乎线性的预期复杂性这个问题。本文讨论了有向图的序贯短截线匹配问题，并证明了这个过程可以被模化为具有渐近等概率的简单有向图的样本。该过程从一个空的边缘集开始，并以一定的状态依赖偏置重复添加边缘，直到满足所需的度序列，同时避免放置双边缘或自循环。我们表明，均匀采样是在稀疏制度，当最大程度的$d_\text{max}$是渐近占主导地位的$m^{1/4}$，其中$m$是边缘的数量。证明是基于导出各种组合的估计与给定的有向度序列的有向图的数量和控制这些估计在大型有向图的浓度。这表明，顺序存根匹配可以被看作是一个实用的算法几乎均匀采样的有向图，我们表明，该算法可以实现功能线性预期运行时$O(m)$。
英文摘要：
Uniform sampling of simple graphs having a given degree sequence is a known problem with exponential complexity in the square of the mean degree. For undirected graphs, randomised approximation algorithms have nonetheless been shown to achieve almost linear expected complexity for this problem. Here we discuss the sequential stub matching for directed graphs and show that this process can be mould to sample simple digraphs with asymptotically equal probability. The process starts with an empty edge set and repeatedly adds edges to it with a certain state-dependent bias until the desired degree sequence is fulfilled, while avoiding placement of a double edge or self loop. We show that uniform sampling is achieved in the sparse regime, when the maximum degree $d_\text{max}$ is asymptotically dominated by $m^{1/4}$, where $m$ is the number of edges. The proof is based on deriving various combinatorial estimates related to the number of digraphs with a given directed degree sequence and controlling concentration of these estimates in large digraphs. This suggests that the sequential stub matching can be viewed as a practical algorithm for almost uniform sampling of digraphs, and we show that this algorithm can be implemented to feature linear expected runtime $O(m)$.
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