Exact bounds for degree centralization

Degree centralization is a simple and widely used index of degree distribution concentration in social networks. Conventionally, the centralization score is normalized by the maximum value attainable at the observed order; this results in a measure on the unit interval. Here, exact bounds are provided for degree centralization given density as well as order. It is shown that roughly half of the region of conceivable degree centralization scores is actually feasible, and that the geometry of the feasible region alters with graph size. Concentration of the distribution of graphs within the feasible region is shown, and a renormalized family of degree centralization indices which adjusts for both density and order is provided.