A working hypothesis that says that the distributions that are from network-related things are power laws; other stuff tends to be LogNormal.
This could be called the graph hypothesis instead.
Of course this is a gross stereotyping of many kinds of distributions and systems but I think there might be something here... Moreover 'network-related' needs a good definition.
For example, the distribution of file sizes on a computer is a LogNormal distribution (see LegoNickHardDiskFileSizes), and their is nothing particularly 'networky' about file sizes. However the number of child directories per directory (see LegoNickNumDirectoryChildren), which corresponds to the out-degree distribution of nodes in the directory tree, and hence is 'network-related', follows a power law.