In this paper we investigated the possibility to use the magnetic Laplacian to characterize directed graphs (a.k.a. networks). Many interesting results are obtained, including the finding that community structure is related to rotational symmetry in the spectral measurements for a type of stochastic block model. Due the hermiticity property of the magnetic Laplacian we show here how to scale our approach to larger networks containing hundreds of thousands of nodes using the Kernel Polynomial Method (KPM). We also propose to combine the KPM with the Wasserstein metric in order to measure distances between networks even when these networks are directed, large and have different sizes, a hard problem which cannot be tackled by previous methods presented in the literature. In addition, our python package is publicly available at \href{https://github.com/stdogpkg/emate}{github.com/stdogpkg/emate}. The codes can run in both CPU and GPU and can estimate the spectral density and related trace functions, such as entropy and Estrada index, even in directed or undirected networks with million of nodes.

中文翻译：

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通过磁拉普拉斯算子的光谱表征和比较大型有向图

在本文中，我们研究了使用磁性拉普拉斯算子来表征有向图（又名网络）的可能性。获得了许多有趣的结果，包括发现群落结构与一类随机块模型的光谱测量中的旋转对称性有关。由于磁性拉普拉斯算子的隐性特性，我们在这里展示了如何使用核多项式方法 (KPM) 将我们的方法扩展到包含数十万个节点的更大网络。我们还建议将 KPM 与 Wasserstein 度量相结合，以测量网络之间的距离，即使这些网络是有向的、大的且具有不同的大小，这是文献中以前的方法无法解决的难题。此外，我们的 Python 包可在 \href{https://github.com 上公开获得。com/stdogpkg/emate}{github.com/stdogpkg/emate}。这些代码可以在 CPU 和 GPU 中运行，甚至可以在具有数百万个节点的有向或无向网络中估计谱密度和相关的跟踪函数，例如熵和 Estrada 指数。