Hi Reuben, yeah in order for Network to be able to work, it needs to organize the curves into 2 separate crossing sets, and those sets should form a grid shape similar to how longitude and latitude lines on a globe can make a grid on a flat map. If you have arrangements that can not be "unwrapped" in to a regular flat grid then it won't work.
There's some more discussion on the type of arrangement that Network needs in some of these previous threads:
And Reuben, the reason why the circle with the four spoke curves touching at the middle works is because it is in fact making a 'grid' Network. It's just wrapped around a tight axis.
