有向网络拉普拉斯算子与随机图模型
Directed Network Laplacians and Random Graph Models
摘要：
我们考虑谱方法，发现隐藏的结构在有向网络。我们通过（a）最小化目标函数和（b）最大化随机图模型的可能性来建立和利用节点重排序之间的连接。我们专注于两个现有的光谱方法，建立和分析拉普拉斯风格的矩阵，通过最小化挫折和营养不一致。这些算法的目的是揭示定向周期和线性层次结构，分别。我们表明，重新排序节点使用这两种算法，或将它们映射到一个指定的格，与新的类有向随机图模型。使用这种随机图设置，我们能够在给定的网络上比较两种算法，并量化哪种结构更有可能存在。我们说明了合成和真实网络的方法，并讨论了实际的实施问题。
英文摘要：
We consider spectral methods that uncover hidden structures in directed networks. We establish and exploit connections between node reordering via (a) minimizing an objective function and (b) maximizing the likelihood of a random graph model. We focus on two existing spectral approaches that build and analyse Laplacian-style matrices via the minimization of frustration and trophic incoherence. These algorithms aim to reveal directed periodic and linear hierarchies, respectively. We show that reordering nodes using the two algorithms, or mapping them onto a specified lattice, is associated with new classes of directed random graph models. Using this random graph setting, we are able to compare the two algorithms on a given network and quantify which structure is more likely to be present. We illustrate the approach on synthetic and real networks, and discuss practical implementation issues.
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