小世界图和结构的压缩与对称性
Compression and Symmetry of Small-World Graphs and Structures
摘要：
出于各种目的，特别是在数据压缩的背景下，可以在三个级别上检查图。它的结构可以描述为图的未标记版本;然后可以添加其结构的标记;最后，给定结构和标记，可以描述标记的内容。确定每个层次上存在的信息量并量化它们之间的依赖程度，需要研究对称性，图形自同构，熵和图形压缩性。本文研究了一类小世界图。这些是几何随机图，其中顶点首先连接到圆上最近的邻居，然后根据距离相关的概率分布连接非邻居对。我们建立了该模型的度分布，并利用它证明了模型在适当的参数范围内的不对称性。然后，我们推导出这些随机图的相关熵和结构熵，与图压缩。
英文摘要：
For various purposes and, in particular, in the context of data compression, a graph can be examined at three levels. Its structure can be described as the unlabeled version of the graph; then the labeling of its structure can be added; and finally, given then structure and labeling, the contents of the labels can be described. Determining the amount of information present at each level and quantifying the degree of dependence between them, requires the study of symmetry, graph automorphism, entropy, and graph compressibility. In this paper, we focus on a class of small-world graphs. These are geometric random graphs where vertices are first connected to their nearest neighbors on a circle and then pairs of non-neighbors are connected according to a distance-dependent probability distribution. We establish the degree distribution of this model, and use it to prove the model's asymmetry in an appropriate range of parameters. Then we derive the relevant entropy and structural entropy of these random graphs, in connection with graph compression.
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