发表机构
Saratoga High School; Department of Mathematics, Williams College（萨拉托加高中; 威廉姆斯学院数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究均匀仿射矩形网格网络中的Braess悖论，给出引发悖论的充要条件，计算相关弦的占比，改进Braess比率上界并证明比率最大化弦的延迟特性。
AI 中文摘要
Braess悖论是指在拥塞网络中添加一条边会导致总通行时间增加的现象。我们在有向矩形网格中研究该悖论，其中每条边的延迟函数均为ℓ(x)=ax+b，新增弦的延迟函数为ℓ₊(x)=cx+d，满足a>0且b,c,d≥0。通过与电网类比，我们对总通行时间的变化进行了界定。随后，我们给出了弦在非负系数取值下引发悖论的充要条件，并计算了所有维度不超过100的网格中此类弦的精确占比。此类弦较为稀少，其占比在长宽比接近2:1时达到最大。接着，我们将已有的Braess比率的4/3上界改进为仅依赖网格维度的上界，该上界在正方形网格上趋近于1.207，在狭长网格上趋近于4/3。最后，我们证明了任何使比率最大化的弦必须具有零延迟。
英文摘要
Braess' Paradox is the phenomenon in which adding an edge to a congestion network increases total travel time. We study the paradox in directed rectangular grids where every edge shares the latency function $\ell(x)=ax+b$ and an added chord has latency $\ell_{*}(x) = cx+d$, where $a > 0$ and $b,c,d \ge 0$. Using an analogy with electrical networks, we bound the change in total travel time. We then give a necessary and sufficient condition for a chord to induce the paradox for some choice of nonnegative coefficients and compute the exact proportion of such chords in all grids with dimensions at most $100$. Such chords are scarce, and the fraction is maximized near an aspect ratio of $2:1$. We then improve the established $4/3$ upper bound on the Braess Ratio to one depending only on the grid dimensions, approaching $1.207$ on squares and $4/3$ on thin grids. Finally, we prove any ratio-maximizing chord must have zero latency.
