发表机构
Racah Institute of Physics, The Hebrew University（雷卡物理研究所，希伯来大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究随机正则图中偏心率分布，推导其闭式解析表达式及平均偏心率、众数、方差的表达式，呈现大规模结构详细视图，结果可作大稀疏网络偏心率计算算法基准，对广播等应用有重要意义。
AI 中文摘要
我们推导了由\(N\)个度为\(c\)的节点组成的随机正则图（RRG）中偏心率分布（DoE）的闭式解析表达式。DoE由尾部分布\(P(E > \ell) \simeq 1 - \exp \left[ - \exp \left( - \frac{ e^{b \ell} - \mu }{\beta} \right) \right]\)给出，其中距离\(\ell\)取整数值，\(b = \ln (c - 1)\)是形状参数，\(\beta = \frac{c - 2}{c} N\)是尺度参数，\(\mu = \frac{c - 2}{c} N \ln N\)是位置参数。通过给出完整分布而非单个特征长度尺度，呈现了大规模结构的详细视图。尽管所有节点度相同，但偏心率有显著变化。我们推导了平均偏心率的闭式表达式\(\langle E \rangle \simeq \frac{\ln N}{\ln (c - 1)} + \frac{\ln \ln N}{\ln (c - 1)} - \frac{ \ln c - \ln (c - 2) }{ \ln (c - 1) } + \frac{1}{2}\)。计算了DoE的众数，其随网络大小呈阶梯状分布，且众数\(E_{\rm mode} ={\rm Round} \left( \langle E \rangle \right)\)。还计算了方差\({\rm Var}(E)\)，其随网络大小\(N\)振荡。本文结果可作为大稀疏网络中偏心率计算算法方法的基准。偏心率在广播和全局传播等实际应用中很重要，网络性能由最长延迟时间决定。
英文摘要
We derive a closed-form analytical expression for the distribution of eccentricities (DoE) in random regular graphs (RRGs) that consist of $N$ nodes of degree $c$. The DoE is given by the tail distribution $P(E > \ell) \simeq 1 - \exp \left[ - \exp \left( - \frac{ e^{b \ell} - μ}β \right) \right]$, where the distance $\ell$ takes integer values, $b = \ln (c-1)$ is the shape parameter, $β= \frac{c-2}{c} N$ is the scale parameter and $μ= \frac{c-2}{c} N \ln N$ is the location parameter. By providing the full distribution rather than a single characteristic length scale, we present a detailed view of the large-scale structure. In spite of the fact that the degrees of all the nodes are the same, their eccentricities exhibit non-trivial variations. We derive a closed-form expression for the mean eccentricity, which is given by $\langle E \rangle \simeq \frac{\ln N}{\ln (c-1)} + \frac{\ln \ln N}{\ln (c-1)} - \frac{ \ln c - \ln (c-2) }{ \ln (c-1) } + \frac{1}{2}$. We calculate the mode of the DoE, which exhibits a staircase profile as a function of the network size. Interestingly, the mode is given by $E_{\rm mode} ={\rm Round} \left( \langle E \rangle \right)$, where ${\rm Round}( x )$ is the nearest integer to $x$. We also calculate the variance ${\rm Var}(E)$ and show that it exhibits oscillations as a function of the network size $N$. The results presented in this paper may serve as benchmarks for algorithmic approaches to eccentricity calculations in large sparse networks. The eccentricities are important in practical applications such as broadcasting and global dissemination, where the network performance is determined by the longest delay times.
Comments25 pages, 7 figures
Journal refJ. Stat. Mech. (2026) 093203
