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• W2120672285 abstract "We study the expected topological properties of Čech and Vietoris–Rips complexes built on random points in ℝ d . We find higher-dimensional analogues of known results for connectivity and component counts for random geometric graphs. However, higher homology H k is not monotone when k>0. In particular, for every k>0, we exhibit two thresholds, one where homology passes from vanishing to nonvanishing, and another where it passes back to vanishing. We give asymptotic formulas for the expectation of the Betti numbers in the sparser regimes, and bounds in the denser regimes." @default.
• W2120672285 created "2016-06-24" @default.
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• W2120672285 date "2011-01-19" @default.
• W2120672285 modified "2023-10-03" @default.
• W2120672285 title "Random Geometric Complexes" @default.
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• W2120672285 doi "https://doi.org/10.1007/s00454-010-9319-3" @default.
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