Editing Scale-free network (section)

====Non-linear preferential attachment====
{{See also|Non-linear preferential attachment}}
The Barabási–Albert model assumes that the probability <math>\Pi(k)</math> that a node attaches to node <math>i</math> is proportional to the [[degree (graph theory)|degree]] <math>k</math> of node <math>i</math>. This assumption involves two hypotheses: first, that <math>\Pi(k)</math> depends on <math>k</math>, in contrast to random graphs in which <math>\Pi(k) = p </math>, and second, that the functional form of <math>\Pi(k)</math> is linear in <math>k</math>.

In non-linear preferential attachment, the form of <math>\Pi(k)</math> is not linear, and recent studies have demonstrated that the degree distribution depends strongly on the shape of the function <math>\Pi(k)</math>

Krapivsky, Redner, and Leyvraz<ref name="Krap"/> demonstrate that the scale-free nature of the network is destroyed for nonlinear preferential attachment. The only case in which the topology of the network is scale free is that in which the preferential attachment is [[asymptotically]] linear, i.e. <math>\Pi(k_i)  \sim a_\infty k_i</math> as <math>k_i \to \infty</math>. In this case the rate equation leads to

: <math> P(k) \sim k^{-\gamma}\text{ with }\gamma = 1 + \frac \mu {a_\infty}.</math>

This way the exponent of the degree distribution can be tuned to any value between 2 and <math>\infty</math>.{{clarify|reason=What is mu?|date=November 2021}}