Compression and Symmetry of Small-World Graphs and Structures

Abstract

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.

Entropy of labeled versus unlabeled networks

“…These observations are directly related to the common confusion between unlabeled graph models and delabeled graph models [39,40]. A delabeled graph model starts with a labeled graph model, generates a labeled graph, and then simply removes the node labels in it.…”

“…We first recall a very simple and general relation between the labeled and delabeled entropies [39,40]. We call the latter the unlabeled entropy below, since unlabeled RGGs are identical to delabeled RGGs.…”

Lossy Compression for Lossless Prediction

“…We directly discard such nuisances from the data to improve compression. Others have used symmetries in X for lossless compression of multisets [21], graphs [56][57][58], or structured images [59][60][61][62][63]. We, instead, use invariance of the tasks Y for lossless prediction.…”