Extracting Persistent Clusters in Dynamic Data via Möbius inversion
Abstract
Identifying and representing clusters in timevarying 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 timevarying network. We first construct a complete summary of the evolution of clusters in a given timevarying 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 timevarying 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 $\mathcal{R}$.
 Publication:

arXiv eprints
 Pub Date:
 December 2017
 DOI:
 10.48550/arXiv.1712.04064
 arXiv:
 arXiv:1712.04064
 Bibcode:
 2017arXiv171204064K
 Keywords:

 Mathematics  Algebraic Topology;
 Computer Science  Computational Geometry
 EPrint:
 54 pages, 18 figures. Extensively rewritten. The focus has moved to new results