范围更新网络中的通用二次度定律
Universal Exponent-Two Degree Laws in Range-Renewal Networks
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中文总结 AI 辅助
该研究针对范围更新网络，推导其极限度分布的二次正则变化定律，结合概率估计与几何方法证明，经模拟验证结论，明确边方向与自环对极限度分布的影响。
中文摘要 AI 辅助
设可数字母表上的无限独立同分布随机变量序列通过连接连续符号并删除重复边来生成图。我们确定该范围更新图的极限度分布的精确尾部和局部渐近性。若有序抽样概率满足\boldsymbol{\u03c0}_k \u2208 \u211b\u0392_{-1/\u03b3}（其中0<\u03b3<1），则有向和无向度尾部分别渐近于\boldsymbol{\u03c0}_k^\u03b3和2^\u03b3\boldsymbol{\u03c0}_k^\u03b3，而对应的局部质量分别渐近于\boldsymbol{\u03c0}_k^\u03b3/k和2^\u03b3\boldsymbol{\u03c0}_k^\u03b3/k。因此，两种极限定律均为指数-2的正则变化，且与\boldsymbol{\u03b3}无关；忽略边方向仅影响主振幅。证明结合了发现时间的无限占用估计、剩余未发现质量以及发现间隔的条件几何表示。一致可积性得出尾部渐近性，而几何平滑论证无需对正则变化尾部求导即可得到局部质量。我们还证明删除自环不会改变极限定律。对归一化齐普夫频率的有限样本模拟验证了该渐近结果。
英文摘要
Let an infinite sequence of independent and identically distributed random variables over a countable alphabet generate a graph by joining consecutive symbols and suppressing repeated edges. We determine the exact tail and local asymptotics of the limiting degree distributions of this range-renewal graph. If the ordered sampling probabilities satisfy \(π_k\in\mathrm{RV}_{-1/γ}\) with \(0<γ<1\), then the directed and undirected degree tails are asymptotic to \(π_k^γ\) and \(2^γπ_k^γ\), respectively, while the corresponding local masses are asymptotic to \(π_k^γ/k\) and \(2^γπ_k^γ/k\). Consequently, both limiting laws are regularly varying with index \(-2\), independent of \(γ\); forgetting edge orientation affects only the leading amplitude. The proof combines infinite-occupancy estimates for discovery times and residual unseen mass with a conditional geometric representation of inter-discovery gaps. Uniform integrability yields the tail asymptotics, whereas a geometric-smoothing argument obtains the local masses without differentiating a regularly varying tail. We also prove that deleting self-loops leaves the limiting laws unchanged. Finite-sample simulations for normalized Zipf frequencies illustrate the asymptotic result.
发表机构
- Fudan University（复旦大学）
机构由 AI 辅助整理，请以论文原文为准。
