Motivated by the problem of optimizing sensor network covers, we generalize the persistent homology of simplicial complexes over a single radial parameter to the context of multiple radial parameters. The persistent homology of so-called multiradial (multi)filtrations is identified as a special case of multidimensional persistence. Specifically, we exhibit that the persistent homology of (multi)filtrations corresponds to both generalized persistence modules of the form [special characters omitted] and (multi)graded modules over a polynomial ring. The stability of persistence barcodes/diagrams of multiradial filtrations is derived, along with explicit bounds associated to perturbations in both radii and vertex position. A strengthening of the Vietoris-Rips lemma of [DSG07, p. 346] to the setting of multiple radial parameters is obtained. We also use the categorical framework of [BDSS15] to show the persistent homology modules of multiradial (multi)filtrations are stable.
|Commitee:||Bell, Gregory, Chhetri, Maya, Smyth, Clifford|
|School:||The University of North Carolina at Greensboro|
|Department:||Mathematics and Statistics|
|School Location:||United States -- North Carolina|
|Source:||MAI 55/06M(E), Masters Abstracts International|
|Keywords:||Cech complex, Multidimensional persistence, Persistent homology, Sensor networks, Stability, Vietoris-rips complex|
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