Your shared network? If you research it for your allies to be able to use your phase nodes? I guess yes, but not intentionally, maybe only when that will be the shortest route to somewhere (but not a carefully planned move like you would do..) never seen them though, i dont like games that last for days, I already win most cases before i research shared network

My preferences are largely the same, except that I prefer Random Vast Single.  I used to play primarily Multi, but found that a larger and more intricate network of planets and wormholes around one star makes for a more interesting game, at least from my perspective.

But I digress.  My main point is that, because of my map preferences, I frequently find myself in a very late-stage game with a Vasari.  Although I have not noticed the AI using a shared network, I'm pretty sure have seen my own trade ships apparently using a Vasari network, so the AI seems to be capable of recognizing and using one when determining the shortest route between two planets. 

The conditions for two AI players to share such a network would have to be that they are allied, correct?  And for you to see it you would also have to have shared ship vision with one or both?  That would seem to be a relatively limited set of conditions, so perhaps it is not that it doesn't happen, but rather that you have just not been in a position to see it.  If that is the case, then the way to increase your odds of would be to select player factions (instead of letting them be determined randomly), put two Vasari on a locked team and work very hard to (1) survive to late game and (2) share ship vision with both.  Then if it happens when they are ganging up on another player, you will be able to see it.

Hmmm...sounds like a lot of work.  Maybe someone from the development team will answer your question.