基于Möbius反演的动态数据持久簇提取
Extracting Persistent Clusters in Dynamic Data via Möbius inversion
摘要：
在研究自然界、移动终端网络或社交网络中出现的集体行为时，识别和表示随时间变化的网络数据中的集群是特别重要的。基于组合论、分类论和持久性理论的观点，我们建立了一个稳定的函子管道，用于总结时变网络中簇的演化。我们首先构建一个完整的总结集群的演化在一个给定的时变网络的一组实体$X$，其中采取形式的formigram。该形式图可以理解为由$X$的子集标记的特定Reeb图$\mathcal{R}$。通过应用莫比乌斯反演的formigram在两种不同的方式，我们得到两个双重概念的图：最大组图和持久性聚类图，这两者都是在一个'注释'条形码的形式。最大群图由相应的最大群标注的时间间隔组成--这是Buchin等人提出的一个概念，这意味着我们承认极大群的概念是Patel的广义持久图的一个特殊例子。另一方面，持久性聚类图主要通过在给定的时变网络中用某些合并/解散事件注释Reeb图$\mathcal{R}$的Z字形条形码中的间隔来获得。我们表明，这两个图是完全不变的formigrams（或等价的轨迹分组结构由Buchin等人）。因此包含比Reeb图$\mathcal{R}$更多的信息。
英文摘要：
Identifying and representing clusters in time-varying network data is of particular importance when studying collective behaviors emerging in nature, in mobile device networks or in social networks. Based on combinatorial, categorical, and persistence theoretic viewpoints, we establish a stable functorial pipeline for the summarization of the evolution of clusters in a time-varying network. We first construct a complete summary of the evolution of clusters in a given time-varying network over a set of entities $X$ of which takes the form of a formigram. This formigram can be understood as a certain Reeb graph $\mathcal{R}$ which is labeled by subsets of $X$. By applying Möbius inversion to the formigram in two different manners, we obtain two dual notions of diagram: the maximal group diagram and the persistence clustergram, both of which are in the form of an `annotated' barcode. The maximal group diagram consists of time intervals annotated by their corresponding maximal groups -- a notion due to Buchin et al., implying that we recognize the notion of maximal groups as a special instance of generalized persistence diagram by Patel. On the other hand, the persistence clustergram is mostly obtained by annotating the intervals in the zigzag barcode of the Reeb graph $\mathcal{R}$ with certain merging/disbanding events in the given time-varying network. We show that both diagrams are complete invariants of formigrams (or equivalently of trajectory grouping structure by Buchin et al.) and thus contain more information than the Reeb graph $\mathcal{R}$.
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