定位偏好聚合冲突：一种使用层的图论方法
Localizing Preference Aggregation Conflicts: A Graph-Theoretic Approach Using Sheaves
AI总结：
该研究利用基于离散层的图论框架定位偏好聚合冲突，通过障碍轨迹识别冲突选民对，形式化不相容指数量化冲突，开发层论前推操作模拟选民合并，展示图商如何将边冲突转化为局部不可能性。
摘要：
我们引入了一个基于离散层的图论框架，用于诊断和定位偏好聚合中的不一致性，更广泛地说，用于定位由许多重叠源提供的部分排名融合中的不一致性。与诸如HodgeRank等将比较嵌入数值流的线性化方法不同，此方法纯粹保持序数性质，并通过障碍轨迹在交互结构中定位冲突，识别哪些选民对不一致。我们形式化了不相容指数以量化这些局部冲突，并使用马洛模型研究它们在随机变化下的行为。我们进一步开发了一种层论前推操作来模拟选民合并，通过多项式时间约束图算法实现。我们证明图商将分布式边冲突转化为局部不可能性（空茎），从拓扑角度展示了聚合悖论如何在不同尺度上持续存在。
英文摘要：
We introduce a graph-theoretic framework based on discrete sheaves to diagnose and localize inconsistencies in preference aggregation and, more broadly, in the fusion of partial rankings supplied by many overlapping sources. Unlike linearization methods such as HodgeRank, which embed comparisons into a numerical flow, this approach stays purely ordinal and locates conflict in the interaction structure via the Obstruction Locus, identifying which voter pairs fail to cohere. We formalize the Incompatibility Index to quantify these local conflicts and examine their behavior under stochastic variations using the Mallows model. We further develop a sheaf-theoretic pushforward operation to model voter merging, implemented via a polynomial-time constraint digraph algorithm. We demonstrate that graph quotients transform distributed edge conflicts into local impossibilities (empty stalks), showing topologically how aggregation paradoxes can persist across scales.
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