Natural graphs that are not scale free It is now a well known observation that many graphical structures that arise in natural settings tend to obey scale-free properties, such as the power law of degree distribution. 

Are there any good example of natural graphs that are fairly random and not do not necessarily obey scale-free properties?
The connections between neurons do not form scale-free networks, I think. See e.g. Figure 1 on p.181 of http://www.indiana.edu/~cortex/Koetter_chapter_modified.pdf and observe that the vertex degrees only span a single order of magnitude.