从旋转拉普拉斯矩阵的谱中寻找有向图和马尔可夫链中的近周期分量
Finding Nearly-Periodic Components in Digraphs and Markov Chains from the Spectrum of Rotated Laplacian Matrices
AI总结：
研究从旋转拉普拉斯矩阵谱中找有向图和马尔可夫链近周期分量，提出并分析谱算法周期比变体，该算法随机多项式时间运行，能找近周期分量，还推出新的高阶切格型不等式及相关概率上界定理。
AI中文摘要：
受有向图谱近似概念最新进展的启发，我们研究通过一类旋转拉普拉斯矩阵的谱来寻找有向图中周期结构的谱算法。这类拉普拉斯矩阵先前由Lange等人研究过。我们考虑了一个推广了Trevisan二分比的周期比概念，并表明它与旋转拉普拉斯矩阵的谱密切相关。对于给定的\(p\in\mathbb{N}\)，如果有向图是强连通的且表示一个马尔可夫链，这个周期比是该马尔可夫链接近具有周期\(p\)的定量度量。我们提出并分析了Louis等人谱算法的周期比变体。我们证明该算法在随机多项式时间内运行，能找到许多近周期分量，这也意味着一个新的高阶切格型不等式。作为分析的一部分，我们证明了一个新定理，它给出了两个坐标相关复高斯随机变量序列最大模在不同索引处出现的概率的上界。之前类似结果仅对实高斯随机变量已知。
英文摘要：
Inspired by recent advances in notions of spectral approximation of digraphs [Ahm+20], we study spectral algorithms for finding periodic structures in digraphs via the spectrum of a class of rotated Laplacian matrices. This class of Laplacian matrices was previously studied by Lange, Liu, Peyerimhoff, and Post [Lan+15]. We consider a notion of periodicity ratio that generalizes the bipartiteness ratio of Trevisan [Tre09], and show that it is closely related to the spectrum of rotated Laplacian matrices. In particular, if the digraph is strongly connected and represents a Markov chain, this periodicity ratio for a given $p \in \mathbb{N}$ is a quantitative measure of how close this Markov chain is to having periodicity $p$. We propose and analyze a periodicity-ratio variant of the spectral algorithm by Louis, Raghavendra, Tetali and Vempala [Lou+12]. We show that the algorithm runs in randomized polynomial time and can find many nearly periodic components (i.e, components with small periodicity ratio). This also implies a new higher-order Cheeger-type inequality for periodicity in the spirit of that in [Lou+12; LOT14]. As part of our analysis, we prove a new theorem that upper bounds the probability that the largest magnitudes of two sequences of coordinate-wise correlated complex Gaussian random variables occur at different indices, which may be of independent interest. Previously, an analogous result was known only for real Gaussian random variables.
