# 树的异速标度律

$\displaystyle{ C_i=cA_i^{\eta} }$

$\displaystyle{ C_i=\sum_{j\in Tree_i}A_j }$

## 链和星状结构

$\displaystyle{ A_i=i }$

$\displaystyle{ C_i=\sum_{j=1}^i A_j=\sum_{j=1}^ij=\frac{j(j+1)}{2}=\frac{A_i(A_i+1)}{2}\sim A_i^2 }$

$\displaystyle{ A_i=N }$

$\displaystyle{ C_i=N+1\times N=2N=2A_i\sim A_i }$

## 生成树的异速标度律

Frank和Murrell等人更进一步地探索了树状结构与异速标度律指数之间的关系[2]。首先，它们构建了一个简单的随机生成树模型，该模型模拟了树的随机生长。该模型有两个参数β和θ，它们控制了整个树的形状。β控制的是与根节点直接相连的节点占总节点数的比例，θ控制的是树的平均深度。

$\displaystyle{ P_{ij}=\frac{t_i^{-\theta}}{\sum_{k\in Tree}t_k^{-\theta}} }$

## 参考文献

1. Garlaschelli, Diego; Caldarelli, Guido; Pietronero, Luciano (2003). "Universal scaling relations in food webs". Nature. 423: 165-168.
2. Frank, F.; Murrell, D. (2005). "A simple explanation for universal scaling rela- tions in food webs". Ecology. 86: 325-3263. line feed character in |title= at position 49 (help)
3. Banavar, J.; Rinaldo, A. (1932). "Size and form in efficient transportation networks". Nature. 399: 130-132. More than one of |last1= and |last= specified (help); More than one of |first1= and |first= specified (help)