Persistent homology in graph power filtrations #MMPMID27853540
Parks AD; Marchette DJ
R Soc Open Sci 2016[Oct]; 3 (10): ä PMID27853540show ga
The persistence of homological features in simplicial complex representations of big datasets in Rn resulting from Vietoris?Rips or ?ech filtrations is commonly used to probe the topological structure of such datasets. In this paper, the notion of homological persistence in simplicial complexes obtained from power filtrations of graphs is introduced. Specifically, the rth complex, r???1, in such a power filtration is the clique complex of the rth power Gr of a simple graph G. Because the graph distance in G is the relevant proximity parameter, unlike a Euclidean filtration of a dataset where regional scale differences can be an issue, persistence in power filtrations provides a scale-free insight into the topology of G. It is shown that for a power filtration of G, the girth of G defines an r range over which the homology of the complexes in the filtration are guaranteed to persist in all dimensions. The role of chordal graphs as trivial homology delimiters in power filtrations is also discussed and the related notions of ?persistent triviality?, ?transient noise? and ?persistent periodicity? in power filtrations are introduced.
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R+Soc+Open+Sci 2016 ; 3 (10): ä
