# 度分布

## 定义

${\displaystyle d(v_{i_{0}})=\sum _{i=i_{0}}e_{i,j}}$

${\displaystyle d_{out}(v_{i_{0}})=\sum _{i=i_{0}}e_{i,j}}$
${\displaystyle d_{in}(v_{i_{0}})=\sum _{j=i_{0}}e_{i,j}}$

${\displaystyle d:\,v_{i}\mapsto d(v_{i})\quad i=1,2,\cdots ,n.}$

${\displaystyle \forall m\in \mathbb {N} ,\,\,P:m\mapsto P(m)={\frac {\operatorname {Card} \{v_{i}\,|\,d(v_{i})=m\}}{n}},}$[1]

${\displaystyle \sum _{m\in \mathbb {N} }P(m)=1.}$

### 随机图顶点的度分布

${\displaystyle P_{i}(k)=\mathbb {P} (d(v_{i})=k).}$

## 例子

${\displaystyle P(m)={\begin{cases}0.3,&m=4\\0.6,&m=3\\0.1,&m=6\\0,&m\neq 3,4,6\end{cases}}}$

${\displaystyle P(m)={\begin{cases}1,&m=n-1\\0,&m\neq n-1\end{cases}}}$

${\displaystyle P(m)={\binom {n-1}{m}}p^{m}(1-p)^{n-1-m}.}$[2]

${\displaystyle P(k)\propto {\frac {1}{k^{\gamma }}}}$

## 参考文献

1. Newman, M. E. J. The structure and function of complex networks. SIAM Review. 2003, 45 (2): 167–256. Bibcode:2003SIAMR..45..167N. arXiv:cond-mat/0303516. doi:10.1137/S003614450342480.[失效連結]
2. M. E. J. Newman, S. H. Strogatz, D. J. Watts. Random Graphs with Arbitrary Degree Distribution and Their Applications. Phys. Rev. E. 2001, 64. doi:10.1103/PhysRevE.64.026118.
3. ^ 《科学美国人》中文版2003年7月. 无尺度网络. 集智集团. [2011-07-04]. （原始内容存档于2012-01-11）.
4. ^ Albert-László Barabási, Réka Albert. Emergence of Scaling in Random Networks (PDF). Science: 509–512. doi:10.1126/science.286.5439.509.