  The structural human connectome (i.e.\ the network of fiber connections in
the brain) can be analyzed at ever finer spatial resolution thanks to advances
in neuroimaging. Here we analyze several large data sets for the human brain
network made available by the Open Connectome Project. We apply statistical
model selection to characterize the degree distributions of graphs containing
up to $\simeq 10^6$ nodes and $\simeq 10^8$ edges. A three-parameter
generalized Weibull (also known as a stretched exponential) distribution is a
good fit to most of the observed degree distributions. For almost all networks,
simple power laws cannot fit the data, but in some cases there is statistical
support for power laws with an exponential cutoff. We also calculate the
topological (graph) dimension $D$ and the small-world coefficient $\sigma$ of
these networks. While $\sigma$ suggests a small-world topology, we found that
$D < 4$ showing that long-distance connections provide only a small correction
to the topology of the embedding three-dimensional space.
