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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 theunlabeled version of the graph; then the labeling of its structure can beadded; and finally, given then structure and labeling, the contents of thelabels can be described. Determining the amount of information present at eachlevel and quantifying the degree of dependence between them, requires the studyof symmetry, graph automorphism, entropy, and graph compressibility. In thispaper, we focus on a class of small-world graphs. These are geometric randomgraphs where vertices are first connected to their nearest neighbors on acircle and then pairs of non-neighbors are connected according to adistance-dependent probability distribution. We establish the degreedistribution of this model, and use it to prove the model's asymmetry in anappropriate range of parameters. Then we derive the relevant entropy andstructural entropy of these random graphs, in connection with graphcompression.

Compression and Symmetry of Small-World Graphs and Structures