The von Neumann graph entropy is a measure of graph complexity based on the Laplacian spectrum. It has recently found applications in various learning tasks driven by networked data. However, it is computational demanding and hard to interpret using simple structural patterns. Due to the close relation between Lapalcian spectrum and degree sequence, we conjecture that the structural information, defined as the Shannon entropy of the normalized degree sequence, might be a good approximation of the von Neumann graph entropy that is both scalable and interpretable. In this work, we thereby study the difference between the structural information and von Neumann graph entropy named as {\em entropy gap}. Based on the knowledge that the degree sequence is majorized by the Laplacian spectrum, we for the first time prove the entropy gap is between $0$ and $\log_2 e$ in any undirected unweighted graphs. Consequently we certify that the structural information is a good approximation of the von Neumann graph entropy that achieves provable accuracy, scalability, and interpretability simultaneously. This approximation is further applied to two entropy-related tasks: network design and graph similarity measure, where novel graph similarity measure and fast algorithms are proposed. Our experimental results on graphs of various scales and types show that the very small entropy gap readily applies to a wide range of graphs and weighted graphs. As an approximation of the von Neumann graph entropy, the structural information is the only one that achieves both high efficiency and high accuracy among the prominent methods. It is at least two orders of magnitude faster than SLaQ with comparable accuracy. Our structural information based methods also exhibit superior performance in two entropy-related tasks.

中文翻译：

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冯·诺伊曼图熵与结构信息之间的相似性：解释，计算和应用

冯·诺依曼图熵是基于拉普拉斯谱的图复杂度的量度。最近，它在网络数据驱动的各种学习任务中发现了应用。然而，这是计算上的要求，并且难以使用简单的结构模式来解释。由于拉普拉斯谱与度数序列之间的密切关系，我们推测结构信息（定义为归一化度数序列的Shannon熵）可能是可扩展且可解释的冯·诺依曼图熵的良好近似。因此，在这项工作中，我们研究了结构信息和被称为{\ em熵间隙}的von Neumann图熵之间的差异。根据度数序列由拉普拉斯频谱进行了主观化的知识，我们首次证明在任何无向非加权图中，熵差距在$ 0 $和$ \ log_2 e $之间。因此，我们证明结构信息是冯·诺依曼图熵的良好近似，可以同时实现可证明的准确性，可伸缩性和可解释性。这种近似进一步应用于与熵有关的两个任务：网络设计和图相似性度量，其中提出了新颖的图相似性度量和快速算法。我们在各种比例和类型的图上的实验结果表明，非常小的熵差距很容易应用于各种图和加权图。作为冯·诺依曼图熵的近似值，结构信息是主要方法中唯一同时实现高效和高精度的信息。它比SLaQ快至少两个数量级，并且具有可比的精度。我们基于结构信息的方法在两个与熵有关的任务中也表现出卓越的性能。