Limit theorems for Betti numbers of random simplicial complexes

By Matthew Kahle, Elizabeth Meckes. 2010

148 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cb4f21bcff0e1c9fa295269e306aacc56d5c0508

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In Computational Theory and Mathematics

In Discrete Mathematics and Combinatorics

In Mathematical Physics

Citations in the corpus

Topological Consistency via Kernel Estimation (2014, background): “Asymptotic and non-asymptotic analysis of consistency and convergence of topological summaries as the number of observations increase has been examined for a variety of geometric objects using a variety of statistical and probabilistic tools [1, 2, 3, 4, 5, 8, 9, 12, 16, 30, 31, 36, 37].”
Random geometric complexes in the thermodynamic regime (2014, background): “Recently [4] has established important – from the point of view of applications – extensions to the results of [3, 18, 20] in which the underlying point process lies on a manifold of lower dimension than an ambient Euclidean space in which the balls of (1.”
Maximally Persistent Cycles in Random Geometric Complexes (2015, background): “cuses mainly on the phase transitions for appearance and vanishing of homology, which can be viewed as higher dimensional generalizations of the phase transition for connectivity in random graphs. In [7, 10, 41, 52] more emphasis was given to the distribution of the Betti numbers, namely the number ”
Clique topology reveals intrinsic geometric structure in neural correlations (2015, background): “This reliability has been theoretically predicted (8, 9) and makes it possible to robustly distinguish random from nonrandom structure in the presence of a monotone nonlinearity (Eq.”
On the topology of random complexes built over stationary point processes. (2012, background): “This has lead, for example, to the Čech and Vietoris–Rips complexes on random Euclidean point sets, studied in papers such as [22, 24], with an extension to the manifold setting in [7].”
Limit theory for point processes in manifolds (2011, background): “The limit theory for the kth Betti number of Vietoris–Rips complexes generated by random points in R is given in [25, 26].”