Symmetry and the Entropy of Small-World Structures and Graphs

Graphical data and the network structures that support them are becoming increasingly common and important in engineering and scientific applications. In the context of data compression, such data can be examined at three levels. The structure of a graph can be described as its unlabeled version; the labeling of this structure can be added; and finally, given then structure and labeling, the contents of the labels can be described. In quantifying the amount of information present at each level, and in examining the relationships between them, the notions of symmetry, graph automorphism, and entropy, arise naturally. In this work we consider a class of small-world 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 distribution. We first determine the degree distribution of this model, and then use it to prove that the model is asymmetric in an appropriate range of parameters. Returning to graph compression, our main results are the computation of the entropy and of the structural entropy of these random graph models.