  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.
