Dual Axis map to the rescue!!
One axis should be the paths, the other should be the 'points' or nodes.
In creating the dual axes, you then have control to color and size nodes and paths separately.
Take a look at attached file, pretty sure this is what you want. (Oh, done and saved in Version 10, so you won't be able to open in 9.x, sorry).
If this correctly answers your question, or is at least helpful, please mark it as such so that others can find it quickly.
Tableau Online Deployment Advisor
MapResponse.twbx 27.3 KB