介数中心性

定义

$\displaystyle{ g(v)= \sum_{s \neq v \neq t}\frac{\sigma_{st}(v)}{\sigma_{st}} }$

$\displaystyle{ \mbox{normal}(g(v)) = \frac{g(v) - \min(g)}{\max(g) - \min(g)} }$

$\displaystyle{ \max(normal) = 1 }$
$\displaystyle{ \min(normal) = 0 }$

真实网络和模型网络中的负荷分配

模型网络

$\displaystyle{ \gamma }$为不同值时，无标度网络中负荷的幂律分布曲线.圆圈： $\displaystyle{ \gamma=2.2 }$，正方形：$\displaystyle{ \gamma=2.5 }$，钻石：$\displaystyle{ \gamma=3.0 }$，X：$\displaystyle{ \gamma=4.0 }$，三角形：$\displaystyle{ \gamma= \infin }$ [1]

$\displaystyle{ P(g) \approx g^{-\delta} }$ (1)

$\displaystyle{ g(k) \approx k^\eta }$.

$\displaystyle{ P(g)= \int P(k) \delta (g-k^\eta) dk }$

$\displaystyle{ P(g\gg1)= \int k^{-\gamma} \delta (g-k^\eta) dk }$
$\displaystyle{ \sim g^{-1-\frac{\gamma -1}{\eta}} }$

$\displaystyle{ \eta=\frac{\gamma -1}{\delta -1} }$

$\displaystyle{ \eta = 2 \rarr \delta = \frac{\gamma +1}{2} }$

< math > eta [/itex] 的最大值(也是 <mat > delta [/itex] 的最小值)为具有 非零分簇Non-vanishing clustering的网络的负载指数设置了界限。

$\displaystyle{ \eta \le 2 \rarr \delta \ge \frac{\gamma +1}{2} }$

加权网络

$\displaystyle{ s_{i} = \sum_{j=1}^{N} a_{ij}w_{ij} }$

$\displaystyle{ a{ij} }$$\displaystyle{ w{ij} }$ 分别作为 $\displaystyle{ i }$$\displaystyle{ j }$ 节点之间的邻接矩阵和权值矩阵。

$\displaystyle{ s(k) \approx k^\beta }$

$\displaystyle{ s(b)\approx b^{\alpha} }$

渗流中心性

$\displaystyle{ PC^t(v)= \frac{1}{N-2}\sum_{s \neq v \neq r}\frac{\sigma_{sr}(v)}{\sigma_{sr}}\frac{{x^t}_s}{{\sum {[{x^t}_i}]}-{x^t}_v} }$

备注

1. K.-I. Goh, B. Kahng, and D. Kim PhysRevLett.87.278701
2. M. Barthélemy. Betweenness centrality in large complex networks. Eur. Phys. J. B 38, 163–168 (2004)
3. Kwang-Il Goh, Eulsik Oh, Hawoong Jeong, Byungnam Kahng, and Doochul Kim. Classification of scale-free networks. PNAS (2002) vol. 99 no. 2
4. A. Barrat, M. Barthelemy, R. Pastor-Satorras, and A. Vespignani. The architecture of complex weighted networks. PNAS (2004) vol. 101 no. 11
5. Piraveenan, Mahendra (2013). "Percolation Centrality: Quantifying Graph-Theoretic Impact of Nodes during Percolation in Networks". PLOS ONE. 8 (1): e53095. Bibcode:2013PLoSO...853095P. doi:10.1371/journal.pone.0053095. PMC 3551907. PMID 23349699.

参考文献

• Freeman, Linton (1977). "A set of measures of centrality based on betweenness". Sociometry. 40 (1): 35–41. doi:10.2307/3033543. JSTOR 3033543.
• Goh, K. I.; Kahng, B.; Kim, D. (2001). "Universal Behavior of Load Distribution in Scale-Free Networks". Physical Review Letters. 87 (27): 278701. arXiv:cond-mat/0106565. Bibcode:2001PhRvL..87A8701G. doi:10.1103/physrevlett.87.278701.
• Mantrach, Amin; et al. (2010). "The sum-over-paths covariance kernel: A novel covariance measure between nodes of a directed graph". IEEE Transactions on Pattern Analysis and Machine Intelligence. 32 (6): 1112–1126. doi:10.1109/tpami.2009.78.
• Moxley, Robert L.; Moxley, Nancy F. (1974). "Determining Point-Centrality in Uncontrived Social Networks". Sociometry. 37 (1): 122–130. doi:10.2307/2786472. JSTOR 2786472.
• Newman, M. E. J. (2010). Networks: An Introduction. Oxford, UK: Oxford University Press. ISBN 978-0199206650.
• Dolev, Shlomi; Elovici, Yuval; Puzis, Rami (2010). "Routing betweenness centrality". J. ACM. 57 (4): 25:1–25:27. doi:10.1145/1734213.1734219.