冯·诺依曼图熵与结构信息的相似性：解释、计算与应用
On the Similarity between von Neumann Graph Entropy and Structural Information: Interpretation, Computation, and Applications
摘要：
冯·诺依曼图熵是基于拉普拉斯谱的图复杂度的度量。它最近在网络数据驱动的各种学习任务中找到了应用。然而，它是计算要求高，很难解释使用简单的结构模式。由于Lapalcian谱和度序列之间的密切关系，我们推测，结构信息，定义为归一化度序列的香农熵，可能是一个很好的近似的冯诺依曼图熵，是可扩展的和可解释的。在这项工作中，我们从而研究之间的差异的结构信息和冯诺依曼图熵命名为{ \em熵差距}。基于度序列被Laplacian谱优化的知识，首次证明了任意无向赋权图的$0$和$\log_2 e$之间存在熵差.因此，我们证明，结构信息是一个很好的近似冯诺依曼图熵，同时实现可证明的准确性，可扩展性和可解释性。这种近似进一步应用到两个熵相关的任务：网络设计和图相似性度量，其中新的图相似性度量和快速算法被提出。我们在不同尺度和类型的图上的实验结果表明，非常小的熵差距很容易适用于广泛的图和加权图。作为冯诺依曼图熵的近似，结构信息是唯一一个实现高效率和高精度的突出方法。它至少比SLaQ快两个数量级，精度相当。我们的结构信息为基础的方法也表现出优越的性能在两个熵相关的任务。
英文摘要：
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.
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