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. We further study two entropy based applications which can benefit from the bounded entropy gap and structural information: network design and graph similarity measure. We combine greedy method and pruning strategy to develop fast algorithm for the network design, and propose a novel graph similarity measure with a fast incremental algorithm for graph streams. 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 based applications.

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