Pure point diffraction implies zero entropy for Delone sets with uniform cluster frequencies

Delone sets of finite local complexity in Euclidean space are investigated. We show that such a set has patch counting and topological entropy 0 if it has uniform cluster frequencies and is pure point diffractive. We also note that the patch counting entropy is 0 whenever the repetitivity function satisfies a certain growth restriction.Comment: 16 pages; revised and slightly expanded versio

Provable and practical approximations for the degree distribution using sublinear graph samples

The degree distribution is one of the most fundamental properties used in the analysis of massive graphs. There is a large literature on graph sampling, where the goal is to estimate properties (especially the degree distribution) of a large graph through a small, random sample. The degree distribution estimation poses a significant challenge, due to its heavy-tailed nature and the large variance in degrees. We design a new algorithm, SADDLES, for this problem, using recent mathematical techniques from the field of sublinear algorithms. The SADDLES algorithm gives provably accurate outputs for all values of the degree distribution. For the analysis, we define two fatness measures of the degree distribution, called the $h$-index and the $z$-index. We prove that SADDLES is sublinear in the graph size when these indices are large. A corollary of this result is a provably sublinear algorithm for any degree distribution bounded below by a power law. We deploy our new algorithm on a variety of real datasets and demonstrate its excellent empirical behavior. In all instances, we get extremely accurate approximations for all values in the degree distribution by observing at most $1\%$ of the vertices. This is a major improvement over the state-of-the-art sampling algorithms, which typically sample more than $10\%$ of the vertices to give comparable results. We also observe that the $h$ and $z$-indices of real graphs are large, validating our theoretical analysis.Comment: Longer version of the WWW 2018 submissio