在加权随机匹配的图表上随机走路的阶段过渡
Zsuzsanna Baran1, Jonathan Hermon2, Anđela Šarković1
1University of Cambridge, Cambridge, UK.
概括
这项研究研究了切断现象在加权的随机走在图表上. 我们建立了边缘权重和图形属性的条件,这些条件对于切线是必要的和足够的,特别是对于具有多项式增长和顶点过渡性的图形.
科学领域:
- 图形理论 图形理论
- 可能性理论概率理论.
- 计算机科学 计算机科学
背景情况:
- 随机走在图形上对于分析图形结构和动态至关重要.
- 切断现象,混合时间的急剧过渡,对于理解随机步行行为至关重要.
- 权重图引入了分析随机步行趋同的复杂性.
研究的目的:
- 为了确定在图表序列上的加权随机走路中存在切线的条件.
- 分析边缘权重 (ε) 和图形属性 (度,增长,过度) 对切线的影响.
- 在一般和特定的图形类 (例如扩展器) 中全面了解切断现象.
主要方法:
- 引入一个修改的图形G*通过添加一个随机的完美匹配与重量ε.
- 对于具有多项式增长的图形,对log(1/ε) 和loggadgadgadgadVngad的关系进行分析.
- 研究顶点过渡图及其对切断界限的必要条件和充分条件的影响.
- 在简单的随机步行中检查线性增长的的图.
- 研究扩展器图形及其对切线的补充机制.
主要成果:
- 对于具有多项式增长的图形,log(1/ε) ≪ loggadgadgadVngad是切线的足够条件.
- 对于顶点过渡的图形,这个条件也是必要的.
- 对于线性增长的的图形,1/ε ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≫ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≫ ≫ ≪ ≪ ≪ ≪ ≪ ≫ ≫ ≪ ≪ ≪ ≪ ≫ ≪ ≪ ≪ ≪ ≫ ≪ ≪ ≪ ≪ ≪ ≫ ≫ ≪ ≫ ≪ ≪ ≪ ≪ ≪ ≪ ≫ ≪ ≫ ≫ ≫ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≪ ≫ ≫ ≪ ≫ ≫ ≪ ≫ ≪ ≫ ≪ ≪ ≪ ≫ ≫ ≪ ≫ ≫ ≪ ≪ ≪ ≫ ≫ ≫ ≪ ≫ ≪ ≫ ≫ ≪ ≫ ≫ ≫ ≪ ≫ ≪ ≫ ≫ ≫ ≫ ≫
- 提供了一个完整的分析为扩展器图表在政权1/ε 记录下来Vn 下.
结论:
- 在加权随机步行中存在切断的存在强烈依赖于边缘权重和图形大小/结构之间的关系.
- 特定的图形属性,如多项式增长和顶点过渡性,显著影响了切线条件.
- 该研究提供了精确的数学标准,用于预测各种图形设置中的切断现象.
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