Editing Scale-free network (section)

====Mediation-driven attachment (MDA) model====

In the [[mediation-driven attachment model|mediation-driven attachment (MDA) model]], a new node coming with <math>m</math> edges picks an existing connected node at random and then connects itself, not with that one, but with <math>m</math> of its neighbors, also chosen at random. The probability <math>\Pi(i)</math> that the node <math>i</math> of the existing node picked is

: <math> \Pi(i) = \frac{k_i} N \frac{\sum_{j=1}^{k_i} \frac 1 {k_j} }{k_i}.</math>

The factor <math>\frac{\sum_{j=1}^{k_i} \frac 1 {k_j} }{k_i}</math> is the inverse of the harmonic mean
(IHM) of degrees of the <math>k_i</math> neighbors of a node <math>i</math>. Extensive numerical investigation suggest that for approximately <math>m> 14</math> the mean IHM value in the large <math>N</math> limit becomes a constant which means <math>\Pi(i) \propto k_i</math>. It implies that the higher the
links (degree) a node has, the higher its chance of gaining more links since they can be
reached in a larger number of ways through mediators which essentially embodies the intuitive
idea of rich get richer mechanism (or the preferential attachment rule of the Barabasi–Albert model). Therefore, the MDA network can be seen to follow
the PA rule but in disguise.<ref>{{cite journal | last1 = Hassan | first1 = M. K. | last2 = Islam | first2 = Liana | last3 = Arefinul Haque | first3 = Syed | year = 2017 | title = Degree distribution, rank-size distribution, and leadership persistence in mediation-driven attachment networks | doi = 10.1016/j.physa.2016.11.001 | journal = Physica A | volume = 469 | pages = 23–30 | arxiv = 1411.3444 | bibcode = 2017PhyA..469...23H | s2cid = 51976352 }}</ref>

However, for <math>m=1</math> it describes the winner takes it all mechanism as we find that almost <math>99\%</math> of the total nodes has degree one and one is super-rich in degree. As <math>m</math> value increases the disparity between the super rich and poor decreases and as <math>m>14</math> we find a transition from rich get super richer to rich get richer mechanism.