as you can see in both pictures, i selected all the lines at once (see the trimming points) and applied the trim command. As second step i started with each region separately for the Network Command. Hope my explanation makes sense.
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.