随机递归树中的秩与层分布
The rank and layer distributions in random recursive trees
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中文总结 AI 辅助
本文推导随机递归树中秩与层分布的精确递推式，证明二者分别呈阶乘、几何衰减且对应节点数自平均，提出不依赖标签或根节点的通用分析框架。
中文摘要 AI 辅助
网络中节点深度的分布对分析网络结构至关重要，秩（指节点到最近叶节点（度为1的节点）的距离）和层（指需剥离所有叶节点的次数，使该节点成为叶节点）是衡量节点在网络中深度的两个指标。本文推导了随机递归树中秩与层分布的精确递推表达式，证明秩分布呈阶乘衰减、层分布呈几何衰减，且具有固定秩或层的节点数具有自平均性。与以往研究不同，本文方法不依赖标签或根节点，为分析复杂网络中的秩与层分布提供了更通用的框架。
英文摘要
The distribution of node depths in a network is crucial for analyzing network structure. Two measures, rank and layer, quantify how deep inside a network a node is. The rank is the node's distance to the closest leaf, a node of degree 1. The layer is the number of times all leaves must be stripped off for the node to become a leaf. We derive exact recursive expressions for the rank and layer distributions in random recursive trees. We show that the rank and layer distributions decay factorially and geometrically, respectively, and prove the self-averaging of the numbers of nodes with fixed rank or layer. Unlike previous studies, our approach does not depend on labels or a root node, providing a more versatile framework for analyzing rank and layer distributions in complex networks.
