title: Persistent homology in graph power filtrations.
abstract: The persistence of homological features in simplicial complex representations of big datasets inRnresulting from Vietoris-Rips or Čech 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, therth complex,r≥ 1, in such a power filtration is the clique complex of therth powerGrof a simple graphG. Because the graph distance inGis 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 ofG. It is shown that for a power filtration ofG, the girth ofGdefines anrrange 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.
authors.affiliation: Electromagnetic and Sensor Systems Department , Naval Surface Warfare Center Dahlgren Division , 18444 Frontage Road Suite 327, Dahlgren, VA 22448-5161 , USA.
authors.forename: Allen D
authors.identifier: 0000-0002-9260-9926
authors.initials: AD
authors.lastname: Parks
authors.affiliation: Electromagnetic and Sensor Systems Department , Naval Surface Warfare Center Dahlgren Division , 18444 Frontage Road Suite 327, Dahlgren, VA 22448-5161 , USA.
authors.forename: David J
authors.initials: DJ
authors.lastname: Marchette
keywords: graph power; graph topology; homology; persistence