Duplicates b/c of Node References, Views "Group By" not helping
Well, as it turns out, I avoided this issue by adding another filter, on the 'list' nodes (the children), and I'm lucky that I have one child 'list' node per network that has a unique flag (a 'master' checkbox field that can only be checked on one child per parent 'network' node).

I filtered to only child 'list' nodes that contain that flag, and this means that my parent 'network' nodes only show once.

I still wish I didn't have to do it this way, though. If anyone else can think of a way to actually get a distinct set of nids/titles without my having to use the 'master' flag on the child nodes, I'll definitely mark that answer correct.