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I need to show things moving between nodes along their connection paths similar to this project. I haven't been able to find any examples of it in cytoscape, but I have used cytoscape in the past and prefer to keep using it for this as well. I would appreciate recommendations on how to approach this problem.

You've got a few options...

1. 
   

   The easiest is the Marquee visual style. It produces a "marching ants" illusion in the direction of directed edges. Simply to the Styles tab in the Control Panel and select the "Marquee" style. In the EDGE tab, you can choose from 3 different Marquee Line Types. You could imagine mapping these 3 line types to 3 categories (or bins) of traffic density, for example. Or you could use color, thickness and/or transparency in combination with a marquee style to represent traffic density. You can see an example here:

   
   
   
   • https://youtu.be/MF0zsxEPoPc?t=44
   
   


2. 
   

   There's also an app for animation! This takes the approach of interpolating any visual style (including position and existence) between any set of key frames you provide. So, for example, you would have a start and finish frame and then CyAnimator would make a movie file for you:

   
   
   
   • http://apps.cytoscape.org/apps/cyanimator
   
   


3. 
   

   And yet another completely different approach: with the scripting capabilities of Cytoscape, you can pretty much do whatever you want. The Unit tests for the RCy3 package, for example, ends up being an almost psychedelic display of data vis potential (and the unit tests aren't even at full coverage, shame). So you could direct your own animations in real time with a bit of scripting in R or Python. Here's the RCy3 unit test demo and links to the scripting libs:

   
   
   
   • 
     https://www.youtube.com/watch?v=IXqbdlUnzUE&t=1s (caution: flashing graphics)
   
   
   • https://bioconductor.org/packages/release/bioc/html/RCy3.html
   
   
   • https://py2cytoscape.readthedocs.io/en/latest/