Plotting metabolic networks?
I'm not sure if they will be useful for your application, but you should look into software used to visualize ecological networks (and maybe also software used for drawing electrical charts). The type of data used there is very similar to what you want to plot, with nodes and links along with metadata for links on e.g. rates or connection strenghts.

I can point you to mangal by ecologist Tim Poisot (blogpost on mangal here), SBGN (Systems Biology Graphical Notation), and there is also the R-package diagram. However, I haven't used them much so I cannot say how useful they will be.