Random Geometric Complexes
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- Kahle, M. Discrete Comput Geom (2011) 45: 553. doi:10.1007/s00454-010-9319-3
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We study the expected topological properties of Čech and Vietoris–Rips complexes built on random points in ℝd. We find higher-dimensional analogues of known results for connectivity and component counts for random geometric graphs. However, higher homology Hk 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.