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.

翻译：von Neumann 图形 entropy 是一个基于 Laplacian 频谱的图形复杂度的量度。 它最近发现由网络数据驱动的各种学习任务中的应用程序。 但是, 它在计算上要求和难以使用简单的结构模式来解释。 由于Lapalcian 频谱和度序列之间的密切关系, 我们推测, 在任何未调整的未调整的图表中, 被定义为 Shannon 和 $\ log_ 2 e 的结构性信息可能是 von Neumann 图形 的精度, 它既可缩放又可解释。 在这项工作中, 我们因此研究结构信息与 von Neumann 图形的精度值之间的差异, 名为 $em- rightpility 的精度, 叫做 rqual- prequal preformal preal preality netroforlity 。 这个精确度的精确度在两个模型上应用了我们最接近的直径的直径的直径直径直径 和直径直径的直径直径的直径数据, 在两个直径图中, 这个直径直径直径直径的直径直径直图中, 和直径直图中, 我们的直对两个直图的直径直径直图中, 直图的精确度, 直对于两个直径直为两个直图, 和直图。