Randomly weighted d-complexes: Minimal spanning acycles and Persistence diagrams

Skraba, P and Thoppe, G and Yogeshwaran, D (2020) Randomly weighted d-complexes: Minimal spanning acycles and Persistence diagrams. In: Electronic Journal of Combinatorics, 27 (2).

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Abstract

A weighted d-complex is a simplicial complex of dimension d in which each face is assigned a real-valued weight. We derive three key results here concern- ing persistence diagrams and minimal spanning acycles (MSAs) of such complexes. First, we establish an equivalence between the MSA face-weights and death times in the persistence diagram. Next, we show a novel stability result for the MSA face-weights which, due to our first result, also holds true for the death and birth times, separately. Our final result concerns a perturbation of a mean-field model of randomly weighted d-complexes. The d-face weights here are perturbations of some i.i.d. distribution while all the lower-dimensional faces have a weight of 0. If the per- turbations decay sufficiently quickly, we show that suitably scaled extremal nearest face-weights, face-weights of the d-MSA, and the associated death times converge to an inhomogeneous Poisson point process. This result completely characterizes the xtremal points of persistence diagrams and MSAs. The point process convergence and the asymptotic equivalence of three point processes are new for any weighted random complex model, including even the non-perturbed case. Lastly, as a conse- quence of our stability result, we show that Frieze�s (3) limit for random minimal spanning trees and the recent extension to random MSAs by Hino and Kanazawa also hold in suitable noisy settings. © The authors. Released under the CC BY license (International 4.0).

Item Type:	Journal Article
Publication:	Electronic Journal of Combinatorics
Publisher:	ELECTRONIC JOURNAL OF COMBINATORICS
Additional Information:	The copyright of this article belongs to ELECTRONIC JOURNAL OF COMBINATORICS
Department/Centre:	Division of Electrical Sciences > Computer Science & Automation
Date Deposited:	22 Jun 2020 09:10
Last Modified:	22 Jun 2020 09:10
URI:	http://eprints.iisc.ac.in/id/eprint/65371

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