[论文解读] Smaller Universes for Uniform Sampling of 0,1-matrices with fixed row and column sums

Annabell Berger, Corrie Jacobien Carstens|arXiv (Cornell University)|Mar 7, 2018

Complex Network Analysis Techniques参考文献 2被引用 3

一句话总结

本文提出一种投影的马尔可夫链方法，通过忽略节点标签，对具有固定行列和的0,1-矩阵实现均匀采样，从而有效采样图的同构类。该方法证明投影链的混合时间至少与原始链一样快，通常显著更快，原因在于状态空间更小且初始分布更优；并引入预处理步骤，实现无需六边形移动的标签图高效采样，显著提升切换链与Curveball链的实际性能。

ABSTRACT

An important problem arising in the study of complex networks, for instance in community detection and motif finding, is the sampling of graphs with fixed degree sequence. The equivalent problem of generating random 0,1 matrices with fixed row and column sums is frequently used as a quantitative tool in ecology. It has however proven very challenging to design sampling algorithms that are both fast and unbiased. This article focusses on Markov chain approaches for sampling, where a close-to-random graph is produced by applying a large number N of small changes to a given graph. Examples are the switch chain and Curveball chain, which are both commonly used by practitioners as they are easy to implement and known to sample unbiased when N is large enough. Within theoretical research, much effort has gone into proving bounds on N. However, existing theoretical bounds are impractically large for most applications while experiments suggest that much fewer steps are needed to obtain a good sample. The contribution of this article is twofold. Firstly it is a step towards better understanding of the discrepancy between experimental observations and theoretically proven bounds. In particular, we argue that while existing Markov chain algorithms run on the set of all labelled graphs with a given degree sequence, node labels are unimportant in practice and are usually ignored in determining experimental bounds. We prove that ignoring node labels corresponds to projecting a Markov chain onto equivalence classes of isomorphic graphs and that the resulting projected Markov chain converges to its stationary distribution at least as fast as the original Markov chain. Often convergence is much faster, as we show in examples, explaining part of the difference between theory and experiments...

研究动机与目标

• 解释在具有固定边缘的0,1-矩阵马尔可夫链采样中，理论混合时间界与实际性能之间差异的原因。
• 通过将忽略节点标签的采样过程建模为图同构类上的马尔可夫链投影，形式化其影响。
• 证明投影链的混合时间至少与原始链一样快，且在许多情况下显著加速，原因在于状态空间减小和初始化改善。
• 开发一种预处理步骤，当与投影链结合时，可在无需切换链或Curveball链中使用六边形移动的情况下，实现标签图的快速、无偏采样。
• 证明所提出的组合方法对所有有向图均能生成遍历链，从而在实践中无需使用复杂移动类型。

提出的方法

• 将原始的有标签图上的马尔可夫链投影到同构图的等价类上，将状态空间减少到同构类型。
• 利用谱性质和特征值包含关系，证明投影链的混合时间上界不超过原始链的混合时间。
• 引入线性时间的预处理步骤，通过随机置换节点标签，实现在投影后对标签图的均匀采样。
• 将投影链应用于同构类，然后利用预处理步骤恢复出一个均匀随机的标签图。
• 通过谱隙分析量化混合时间的改进，表明在某些图族上，投影切换链的谱隙为 $ \frac{2}{n} $。
• 证明对于特定度序列，投影链的状态空间从指数级减少到常数大小，例如当 $ l $-正则次级节点时，从 $ \binom{2l}{l} $ 减少到1。

实验结果

研究问题

• RQ1 将马尔可夫链投影到图的同构类上，是否能保持或改善相比原始有标签链的混合时间？
• RQ2 忽略节点标签在多大程度上解释了实践中观察到的比理论界更快的收敛速度？
• RQ3 是否可以使用预处理步骤在投影链之后高效采样标签图，而无需使用如六边形移动等复杂移动？
• RQ4 状态空间缩减对投影链的谱隙和混合时间有何影响？
• RQ5 是否存在某些度序列，使得投影链产生常数大小的状态空间，从而实现在近似瞬间完成采样？

主要发现

• 投影马尔可夫链的收敛速度至少与原始链一样快，其混合时间上界不超过原始链，原因在于特征值包含关系及状态空间的减小。
• 对于具有两个度为 $ n-1 $ 的主节点和 $ n $ 个度为1的次级节点的图族，投影切换链的谱隙为 $ \frac{2}{n} $，混合时间为 $ O(n \log(n/\epsilon)) $。
• 在两个主节点度为 $ l $ 和 $ 2l $ 个度为1的次级节点的情况下，投影链将状态空间从 $ \binom{2l}{l} $ 减少到单个状态，使得在预处理后可立即实现均匀采样。
• 将投影链与线性时间预处理步骤结合，可对所有有向图生成遍历链，从而在切换链与Curveball链中无需使用六边形移动。
• 当原始状态空间具有高度对称性时，投影带来的加速最为显著，因为状态空间大小的减小和初始分布的改善共同导致更快收敛。
• 原始链的混合时间理论界通常不切实际地庞大，但投影链通过聚焦于拓扑结构而非标签，为实现更快、无偏采样提供了实用路径。