If the network is growing and new nodes tend to connect to existing ones with linear probability in the degree of the existing nodes then preferential attachment leads to a scale-free network.
According to Kunegis, Blattner, and Moser several online networks follow a non-linear preferential attachment model.
[1] The co-author network among scientists also shows the signs of sub-linear preferential attachment.
This is believed to be more flexible, and allows the discovery of non-log-linearity of preferential attachment in real networks.
For α > 2 this process happens more extremely, the number of connections between other nodes is still finite in the limit when n goes to infinity.