超图重叠聚类方案的分类
A classification of overlapping clustering schemes for hypergraphs
AI总结：
本文沿函子分类路线研究超图重叠聚类，将可表示性扩展到重叠情形，证明可表示性与切除性、函子性等价，并给出有界扩张图上的多项式时间可计算性结果。
摘要：
图中的社区检测是一个只要出现网络数据就可能相关的问题，因此该问题受到广泛关注，并应用了许多不同的方法和算法。然而，这些方法中有许多难以在理论上进行研究，而且它们优化的目标也有所不同。对该问题及可能方法给出一个通用且严谨的阐述仍然难以实现。我们研究寻找超图重叠聚类的问题，延续 Carlsson 和 Mémoli（2013）开创的将聚类方案分类为函子的研究路线。我们将他们的可表示性概念扩展到重叠情形，证明任何可表示的重叠聚类方案都是切除的和函子的，并且任何切除且函子的聚类方案都同构于一个可表示的聚类方案。我们还注意到，对于简单图，任何可表示的聚类方案在有界扩张图上都可在多项式时间内计算，其指数由表示集中图的最大独立数决定。该结果也适用于非重叠的可表示聚类方案，因此可能具有独立的研究价值。
英文摘要：
Community detection in graphs is a problem that is likely to be relevant whenever network data appears, and consequently the problem has received much attention with many different methods and algorithms applied. However, many of these methods are hard to study theoretically, and they optimise for somewhat different goals. A general and rigorous account of the problem and possible methods remains elusive. We study the problem of finding overlapping clusterings of hypergraphs, continuing the line of research started by Carlsson and Mémoli (2013) of classifying clustering schemes as functors. We extend their notion of representability to the overlapping case, showing that any representable overlapping clustering scheme is excisive and functorial, and any excisive and functorial clustering scheme is isomorphic to a representable clustering scheme. We also note that, for simple graphs, any representable clustering scheme is computable in polynomial time on graphs of bounded expansion, with an exponent determined by the maximum independence number of a graph in the representing set. This result also applies to non-overlapping representable clustering schemes, and so may be of independent interest.
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