Editing Scale-free network (section)

== The scale-free metric ==
On a theoretical level, refinements to the abstract definition of scale-free have been proposed. For example, Li et al. (2005) offered a potentially more precise "scale-free metric". Briefly, let ''G'' be a graph with edge set ''E'', and denote the degree of a vertex <math>v</math> (that is, the number of edges incident to <math>v</math>) by <math>\deg(v)</math>. Define
: <math>s(G) = \sum_{(u,v) \in E} \deg(u) \cdot \deg(v).</math>

This is maximized when high-degree nodes are connected to other high-degree nodes. Now define

: <math>S(G) = \frac{s(G)}{s_\max},</math>

where ''s''<sub>max</sub> is the maximum value of ''s''(''H'') for ''H'' in the set of all graphs with degree distribution identical to that of&nbsp;''G''. This gives a metric between 0 and 1, where a graph ''G'' with small ''S''(''G'') is "scale-rich", and a graph ''G'' with ''S''(''G'') close to 1 is "scale-free". This definition captures the notion of [[self-similarity]] implied in the name "scale-free".