A scale-free network is a connected graph (a network), whose number of links is originating by a starting node and subjects to a power law distribution.
Links between nodes has a particularity: the probability to have a new link in a node is proportional to the existing links. For example, a friends-graph (a connected graph showing the friendship in a group of people) is a SFN, in fact who has more friends has more probability to gain a new friend too.
The graph generated from Erdos Numbers is a Scale-Free Network. Lower EN you have, higher number of annual publication you have.
As far as friends-graph is concerned in 1976 Milgrad made an experiment showing that the distance between two random nodes (not linked each other) is around 6. In 2001 Duncan Watts, making a similar experiment using internet (so with a lot of statistics), confirmed the result.
So, reaching an EN is not too complicated… indeed you may already have one.