As per the definition of power law, the fraction P(k) of nodes with k degree for large values of k , given byP(k) ~k ^-r .In this definition, the term large value is not clearly defined. Does large implies  100 or 10^3 or 10^6? Does the definition implies that vertices in power law graphs have almost constant degree with only few exceptions called hub nodes which have very degree may be 100times higher than other vertices?If the above statement is true, then why do the degree distribution graph for powerlaw graphs do not have only few points with high value of degree and other nodes at the end of disconnected curve compared to close points on decaying curve as in this plot yahoo_web_graph  Is this definition enough to get the number of hub vertices present in the graph or fraction of edges incident on them, given the number of vertices in the graph?
Powerlaw graphs : Number of hubs and fraction of edges incident on them
graph theory;graphs