Random Geometric Complexes

We study the expected topological properties of ˇCech 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. The main technical contribution of the article is the application of discrete Morse theory in geometric probability.

• Type: Journal
• Year: 2011
• Where: DCG
• Authors: Matthew Kahle
• Keywords: Computer Graphics, DCG 2011, Discrete Morse Theory, Main Technical Contribution, Random Geometric Graphs