I got all 3 achievements and 100%.. here is my findings:
Generally the agent moves "closer" to an exit but if there are several options I'm not 100% sure how it decides.
I think it might be trying to also minimize the distance from the other nodes perhaps? anyways it is clear that the agents agenda is actually not to use up all your links but to just move around in some pattern.
My algorithm did:
step 1: (using a simple Queue) calculate distance of each node from exit.
It is important that you keep updating your network as you remove nodes, and this is why you need to redo this every step.
My simple algorithm looked at all the nodes next to the agent and remove the link to the one with the lowest distance from any exit.
This is a simple algorithm and I don't think this is optimal for bigger networks. But it was good enough for this problem.