FFXIV automatically kicked me from Novice Network

[Lazaruz]

think you can manually be re-invited by a mentor.

If they were still a "sprout" a.k.a Novice, then yes.
But the system automaticly removing someone from the Network is a immediate tell that the user is no longer considered a "Novice" by the system

InkstainedGwyn said it best

I absolutely understand wanting to keep in touch with people, but the NN was never meant to be permanent and the game gives many other options that are permanent so that you can continue to talk to the people you've grown close to.