Horton Law in Self-Similar Trees
Horton Law in Self-Similar Trees
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自相似树中的霍顿定律
DOI：
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发表时间：
2015
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影响因子：
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通讯作者：
I. Zaliapin
中科院分区：
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作者：
Yevgeniy Kovchegov;I. Zaliapin
随机树的自相似性与修剪操作有关。修剪$R$会截断树叶及其父边，并从有限树中删除生成的2阶节点链。顶点$v$及其父边的Horton-Strahler序被定义为消除以$v$为根的子树所需的最小剪枝次数。分支是一组相同顺序的相邻顶点和边。Horton数$N_k[K]$和$N_{ij}[K]$被定义为在一个$K$阶有限树中，分别有$k$阶的期望分支数和合并了$j$分支、$j&>i$阶的阶$i$分支的期望数。Tokunaga系数定义为：$T_{ij}[K]=N_{ij}[K]/N_j[K]$。修剪将树顶点的顺序统一减少。一个有根全二叉树称为均值自相似的，如果它的Tokunaga系数关于剪枝是不变的：$T_k：=T_{i，i+k}[K]$。我们证明了对于自相似树，条件$limsup(T_K)^{1/k}0$且每个$kgeq 1$.这项工作是为霍顿定律提供严格基础的一步，霍顿定律在自然分支系统中无处不在，到目前为止还没有得到正式的解释。
Self-similarity of random trees is related to the operation of pruning. Pruning $R$ cuts the leaves and their parental edges and removes the resulting chains of degree-two nodes from a finite tree. A Horton-Strahler order of a vertex $v$ and its parental edge is defined as the minimal number of prunings necessary to eliminate the subtree rooted at $v$. A branch is a group of neighboring vertices and edges of the same order. The Horton numbers $N_k[K]$ and $N_{ij}[K]$ are defined as the expected number of branches of order $k$, and the expected number of order-$i$ branches that merged order-$j$ branches, $j>i$, respectively, in a finite tree of order $K$. The Tokunaga coefficients are defined as $T_{ij}[K]=N_{ij}[K]/N_j[K]$. The pruning decreases the orders of tree vertices by unity. A rooted full binary tree is said to be mean-self-similar if its Tokunaga coefficients are invariant with respect to pruning: $T_k:=T_{i,i+k}[K]$. We show that for self-similar trees, the condition $limsup(T_k)^{1/k} 0$ and every $kgeq 1$. This work is a step toward providing rigorous foundations for the Horton law that, being omnipresent in natural branching systems, has escaped so far a formal explanation.