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Excess degree distribution is the probability distribution, for a node reached by following an edge, of the number of other edges attached to that node. In other words, it is the distribution of outgoing links from a node reached by following a link.

Excess degree distribution is the probability distribution, for a node reached by following an edge, of the number of other edges attached to that node. In other words, it is the distribution of outgoing links from a node reached by following a link.

'''<font color="#ff8000">超额度分布 Excess Degree Distribution</font>'''是通过跟随一条边到达该节点的其他边的数量的概率分布。换句话说，它是通过跟随一个链接从一个节点到达的其传出链接的分布。

'''<font color="#ff8000">超额度分布 Excess Degree Distribution</font>'''的定义是：沿着一条边到达该节点的其他边的数量的概率分布。换句话说，它是通过跟随链接从一个节点到达的其传出链接的分布。

假设一个网络具有度分布<math>

假设一个网络具有度分布<math>

P(k)

P(k)

</math>,通过选择一个节点(随机或非随机)跟踪它的一个邻居(假设至少有一个邻居) ，那么该节点具有<math>

</math>,通过选择一个节点(随机或非随机)跟随它的一个邻近点(假设至少有一个邻近点），那么该节点具有<math>

k

k

</math> 个邻居的概率不是由<math>

</math> 个邻近点的概率不是由<math>

P(k)

P(k)

</math>.给出的原因在于，无论何时在异质网络中选择某个节点，它都更有可能通过跟随该节点的某个现有邻居到达枢纽节点。这些节点具有度<math>

</math>.给出的。造成这一结果的原因在于，无论何时在异质网络中选择某个节点，它都更有可能通过跟随该节点的某个现有邻点到达枢纽节点。这些节点具有度<math>

k

k

</math>的真实概率是<math>

</math>的真实概率是<math>

</math> is the mean-degree (average degree) of the model. It follows to that fact that the average degree of the neighbor of any node is greater than the average degree of that node. In social networks, it mean that your friends, on average, have more friends than you. This is famous as the friendship paradox. It can be shown that a network can have a giant component, if its average excess degree is larger than one:

</math> is the mean-degree (average degree) of the model. It follows to that fact that the average degree of the neighbor of any node is greater than the average degree of that node. In social networks, it mean that your friends, on average, have more friends than you. This is famous as the friendship paradox. It can be shown that a network can have a giant component, if its average excess degree is larger than one:

这里<math>{\langle k \rangle}</math > 是模型的平均度。由此可知，任何节点的邻居的平均度大于该节点的平均度。在社交网络中，这意味着你的朋友平均比你拥有更多的朋友。这就是著名的'''<font color="#ff8000">友谊悖论 Friendship Paradox</font>'''。可以证明，如果一个网络的平均超额度大于1，那么它可以有一个巨大的联通子网络:

这里<math>{\langle k \rangle}</math > 是模型的平均度。由此可知，任何节点的邻近点的平均度大于该节点的平均度。推广到在社交网络络中，这意味着你的朋友平均比你拥有更多的朋友。这就是著名的'''<font color="#ff8000">友谊悖论 Friendship Paradox</font>'''。可以证明，如果一个网络的平均超额度大于1，那么它可以有一个巨大的联通子网络:

Bear in mind that the last two equations are just for the configuration model and to derive the excess degree distribution of a real-word network, we should also add degree correlations into account.

Bear in mind that the last two equations are just for the configuration model and to derive the excess degree distribution of a real-word network, we should also add degree correlations into account.

== The Generating Functions Method ==

== The Generating Functions Method ==