Every link will be a road.
Road building is cheap.
The realistic solution to this problem is to have roads automatically decay by a certain percentage every month. So only roads which someone has an interest in sustaining will remain. The rest will be gone after a year or two. That it may be quicker to build a road and then travel the way than simply to travel is an obvious mockery, but one that is easily fixed by putting a lower limit of twice or thrice the travel time on road construction. That is a small addition and seems perfectly realistic.