From this Wikipedia page, I gather that when the degree distribution of a graph obeys the power law, the graph is termed 'scale-free'. I would like to know the reason for this term.
What has scaling got to do with power-law distributions? What is the precise definition for a scale-free network, in terms of network primitives?
The term Scale-free is many times (ab)used to refer to slightly different graphs. The working definition is to call any graph with an asymptotic power law distribution as scale-free. To understand the reason, consider an ER graph. The degree distribution of this graph follows a Poisson distribution and so it is highly peaked at value $<k>$. Hence in this case, you can say that a "typical vertex" in ER graph looks like a vertex of degree $<k>$. This cannot be true when the distribution is of power-law type since a complete hierarchy of degree values exists in practice and so very large degree vertices coexist along with many small degree vertices. In this case, it would completely wrong to say that a typical vertex in this graph looks like a vertex with degree $<k>$. In other words, there is no "scale" in the graph when the distribution follows a power-law and hence the name.