# 集聚系数

## 整体集聚系数

$L(i) = \left\{ v_j : e_{ij} \in E \and e_{ji} \in E \right\}$

L(i) 里的边的数量就是顶点 $v_i$ 的度，记作 $k_i$$k_i = |L(i)|$

$C_{total}(G) = \frac{3 \times G_{\triangle} }{3 \times G_{\triangle} + G_{\and} }$

$C_{total}(G) = \frac{3 \times G_{\triangle} }{\sum_{i=1}^n \binom{k_i}{ 2}}$[5]

## 局部集聚系数

$C(i) = \frac{2 \Big | \Big \{ e_{jk} : v_j,v_k \in L(i), e_{jk} \in E \Big \} \Big | }{k_i(k_i-1)} .$

$C(i) = \frac{ \Big | \Big \{ e_{jk} : v_j,v_k \in L(i), e_{jk} \in E \Big \} \Big | }{k_i(k_i-1)}.$

$C(i) = \frac{\lambda_G(v_i)}{\tau_G(v_i) + \lambda_G(v_i)}.$

$\tau_G(v_i) + \lambda_G(v_i) = C({k_i},2) = \frac{1}{2}k_i(k_i-1).$

## 平均集聚系数

$\bar{C} = \frac{1}{n}\sum_{i=1}^{n} C(i).$

$\bar{C} = \frac{1}{n}\sum_{i=1}^{n} C(i) = \frac{1}{n}\sum_{i=1}^{n} \frac{\lambda_G(v_i)}{\tau_G(v_i) + \lambda_G(v_i)}$
$C_{total}(G) = \frac{\sum_{i=1}^{n} \lambda_G(v_i)}{\sum_{i=1}^{n}\left( \tau_G(v_i) + \lambda_G(v_i) \right)}$

## 参考来源

1. ^ 王冰、修志龙、唐焕文. 基于复杂网络理论的代谢网络结构研究进展. 《中国生物工程杂志》. 2005 No.6, 25–3: 10–14.
2. ^ P. W. Holland and S. Leinhardt. Transitivity in structural models of small groups. Comparative Group Studies. 1971, 2: 107–124.
3. ^ 3.0 3.1 3.2 D. J. Watts and Steven Strogatz. Collective dynamics of 'small-world' networks. Nature. 1998-06, 393 (6684): 440–442. doi:10.1038/30918. PMID 9623998.
4. ^ R. D. Luce and A. D. Perry. A method of matrix analysis of group structure. Psychometrika. 1949, 14 (1): 95–116. doi:10.1007/BF02289146. PMID 18152948.
5. ^ N. Eggemann and S.D. Noble. The clustering coefficient of a scale-free random graph. Discrete Applied Mathematics. 2009-November 3., 159 (10): 953–965. doi:10.1016/j.dam.2011.02.003.
6. ^ 章忠志、荣莉莉、周涛. 一类无标度合作网络的演化模型. 《系统工程理论与实践》. 2005-11, 11: 55–60.
7. ^ A. Barrat and M. Barthelemy and R. Pastor-Satorras and A. Vespignani. The architecture of complex weighted networks. Proceedings of the National Academy of Sciences. 2004, 101 (11): 3747–3752. doi:10.1073/pnas.0400087101. PMC 374315. PMID 15007165.
8. ^ M. Latapy and C. Magnien and N. Del Vecchio. Basic Notions for the Analysis of Large Two-mode Networks. Social Networks. 2008, 30 (1): 31–48. doi:10.1016/j.socnet.2007.04.006.