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→中介驱动依附(MDA)模型
注意，平均路径长度随N的变化和半径相同。
注意，平均路径长度随N的变化和半径相同。
===中介驱动依附(MDA)模型===
In the [[mediation-driven attachment model|mediation-driven attachment (MDA) model]] in which 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 chosen also 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>
在[[中介驱动依附模型|中介驱动依附(MDA)模型]]中，一个带有<math>m</math>边的新节点随机选择一个已连接的节点，然后不与那个节点连接，而是随机地与它的<math>m</math>个邻居连接。随机选择已连接节点的概率<math>\Pi(i)</math> 为
在[[中介驱动依附模型|中介驱动依附(MDA)模型]]中，一个带有<math>m</math>边的新节点随机选择一个已连接的节点，然后不与那个节点连接，而是随机地与它的<math>m</math>个邻居连接。随机选择已连接节点的概率<math>\Pi(i)</math> 为
: <math> \Pi(i) = \frac{k_i} N \frac{ \sum_{j=1}^{k_i} \frac 1 {k_j} }{k_i}.</math>
: <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 an 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 | date = March 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 | issue = | pages = 23–30 | arxiv = 1411.3444 | bibcode = 2017PhyA..469...23H }}</ref>
因子 <math>\frac{\sum_{j=1}^{k_i}{\frac{1}{k_j}}}{k_i}</math>是节点 <math>i</math>的<math>k_i</math>个相邻节点的度的调和平均（IHM）的逆。大量的数值研究表明，对于<math>m> 14</math>，IHM的值在大<math>N</math>极限下会变成常数，这意味着<math>\Pi(i) \propto k_i</math>。这表明一个节点的度越高，他就有更大的机会获得更多的连接，因为它们可以通过中介以更多的方式达到，这本质上体现了富人越富机制的直观思想（或者BA模型优先链接的规则）。因此，可以看出MDA网络遵循优先链接规则，但是是伪装的。<ref>{{cite journal | last1 = Hassan | first1 = M. K. | last2 = Islam | first2 = Liana | last3 = Arefinul Haque | first3 = Syed | date = March 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 | issue = | pages = 23–30 | arxiv = 1411.3444 | bibcode = 2017PhyA..469...23H }}</ref>
因子 <math>\frac{\sum_{j=1}^{k_i}{\frac{1}{k_j}}}{k_i}</math>是节点 <math>i</math>的<math>k_i</math>个相邻节点的度的调和平均（IHM）的逆。大量的数值研究表明，对于<math>m> 14</math>，IHM的值在大<math>N</math>极限下会变成常数，这意味着<math>\Pi(i) \propto k_i</math>。这表明一个节点的度越高，他就有更大的机会获得更多的连接，因为它们可以通过中介以更多的方式达到，这本质上体现了富人越富机制的直观思想（或者BA模型优先链接的规则）。因此，可以看出MDA网络遵循优先链接规则，但是是伪装的。<ref>{{cite journal | last1 = Hassan | first1 = M. K. | last2 = Islam | first2 = Liana | last3 = Arefinul Haque | first3 = Syed | date = March 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 | issue = | pages = 23–30 | arxiv = 1411.3444 | bibcode = 2017PhyA..469...23H }}</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 have 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.
然而对于<math>m=1</math>，它表现了赢者通吃的机制，因为可以发现几乎<math>99\%</math> 的节点的度仅为1，而某一个节点的度超级大。当<math>m</math>的值增加时，超级富人和穷人之间的差距减小;当<math>m>14</math>时，我们发现了从赢者通吃机制到富人更富机制的转变。
然而对于<math>m=1</math>，它表现了赢者通吃的机制，因为可以发现几乎<math>99\%</math> 的节点的度仅为1，而某一个节点的度超级大。当<math>m</math>的值增加时，超级富人和穷人之间的差距减小;当<math>m>14</math>时，我们发现了从赢者通吃机制到富人更富机制的转变。