Geodesic
The Shadow knows: Empirical Distributions of Minimum Spanning Acycles and Persistence Diagrams of Random Complexes
Discrete analysis (2023) Cet article a éte moissonné depuis la source Scholastica

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In 1985, Frieze showed that the expected sum of the edge weights of the minimum spanning tree (MST) in the uniformly weighted graph converges to $ζ(3)$. Recently, Hino and Kanazawa extended this result to a uniformly weighted simplicial complex, where the role of the MST is played by its higher-dimensional analog -- the Minimum Spanning Acycle (MSA). Our work goes beyond and describes the histogram of all the weights in this random MST and random MSA. Specifically, we show that their empirical distributions converge to a measure based on a concept called the shadow. The shadow of a graph is the set of all the missing transitive edges, and, for a simplicial complex, it is a related topological generalization. As a corollary, we obtain a similar claim for the death times in the persistence diagram corresponding to the above-weighted complex, a result of interest in applied topology.
Publié le : 
@article{DAS_2023_a20,
     author = {Nicolas Fraiman and Sayan Mukherjee and Gugan Thoppe},
     title = {The {Shadow} knows: {Empirical} {Distributions} of {Minimum} {Spanning} {Acycles} and {Persistence} {Diagrams} of {Random} {Complexes}},
     journal = {Discrete analysis},
     year = {2023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DAS_2023_a20/}
}
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Nicolas Fraiman; Sayan Mukherjee; Gugan Thoppe. The Shadow knows: Empirical Distributions of Minimum Spanning Acycles and Persistence Diagrams of Random Complexes. Discrete analysis (2023). http://geodesic.mathdoc.fr/item/DAS_2023_a20/