Abstract
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 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.
Article PDF
Similar content being viewed by others
References
Alon, N., Spencer, J.H.: The Probabilistic Method, Wiley-Interscience Series in Discrete Mathematics and Optimization, 3rd edn. Wiley, Hoboken (2008). With an appendix on the life and work of Paul Erdős
Babson, E., Hoffman, C., Kahle, M.: The fundamental group of random 2-complexes. J. Am. Math. Soc. 24(1), 1–28 (2011)
Barishnokov, Y.: Quantum foam, August 2009. (Talk given at AIM Workshop on “Topological complexity of random sets”)
Björner, A.: Topological methods. In: Handbook of Combinatorics, vol. 2, pp. 1819–1872. Elsevier, Amsterdam (1995)
Bollobás, B., Riordan, O.: Clique percolation. Random Struct. Algorithms 35(3), 294–322 (2009)
Bubenik, P., Carlson, G., Kim, P.T., Luo, Z.M.: Statistical topology via Morse theory, persistence and nonparametric estimation. Algebr. Methods Stat. Probab. II 516, 75 (2010)
Bubenik, P., Kim, P.T.: A statistical approach to persistent homology. Homology Homotopy Appl. 9(2), 337–362 (2007)
Carlsson, G.: Topology and data. Bull. Am. Math. Soc. (N.S.) 46(2), 255–308 (2009)
Chambers, E.W., de Silva, V., Erickson, J., Ghrist, R.: Vietoris–Rips complexes of planar point sets. Discrete Comput. Geom. 44(1), 75–90 (2010)
Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discrete Comput. Geom. 37(1), 103–120 (2007)
Diaconis, P.: Application of topology. Quicktime video, September 2006. (Talk given at MSRI Workshop on “Application of topology in science and engineering,” Quicktime video available on MSRI webpage)
Edelsbrunner, H., Harer, J.: Persistent homology—a survey. In: Surveys on Discrete and Computational Geometry. Contemp. Math., vol. 453, pp. 257–282. Am. Math. Soc., Providence (2008)
Forman, R.: A user’s guide to discrete Morse theory. Sémin. Lothar. Comb. 48, Art. B48c (2002), 35 pp. (electronic)
Gromov, M.: Hyperbolic groups. In: Essays in Group Theory. Math. Sci. Res. Inst. Publ., vol. 8, pp. 75–263. Springer, New York (1987)
Gromov, M.: Random walk in random groups. Geom. Funct. Anal. 13(1), 73–146 (2003)
Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)
Hausmann, J.-C.: On the Vietoris–Rips complexes and a cohomology theory for metric spaces. In: Prospects in Topology, Princeton, NJ, 1994. Ann. of Math. Stud., vol. 138, pp. 175–188. Princeton University Press, Princeton (1995)
Kahle, M.: The neighborhood complex of a random graph. J. Comb. Theory, Ser. A 114(2), 380–387 (2007)
Kahle, M.: Topology of random clique complexes. Discrete Math. 309(6), 1658–1671 (2009)
Kahle, M., Meckes, E.: Limit theorems for Betti numbers of random simplicial complexes. Submitted, arXiv:1009.4130 (2010)
Linial, N., Meshulam, R.: Homological connectivity of random 2-complexes. Combinatorica 26(4), 475–487 (2006)
Linial, N., Novik, I.: How neighborly can a centrally symmetric polytope be? Discrete Comput. Geom. 36(2), 273–281 (2006)
Meshulam, R., Wallach, N.: Homological connectivity of random k-dimensional complexes. Random Struct. Algorithms 34(3), 408–417 (2009)
Niyogi, P., Smale, S., Weinberger, S.: Finding the homology of submanifolds with high confidence from random samples. Discrete Comput. Geom. 39(1–3), 419–441 (2008)
Niyogi, P., Smale, S., Weinberger, S.: A topological view of unsupervised learning from noisy data (2010, to appear)
Penrose, M.: Random Geometric Graphs. Oxford Studies in Probability, vol. 5. Oxford University Press, Oxford (2003)
Pippenger, N., Schleich, K.: Topological characteristics of random triangulated surfaces. Random Struct. Algorithms 28(3), 247–288 (2006)
Vietoris, L.: Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen. Math. Ann. 97(1), 454–472 (1927)
Zomorodian, A., Carlsson, G.: Computing persistent homology. Discrete Comput. Geom. 33(2), 249–274 (2005)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported in part by Stanford’s NSF-RTG grant in geometry & topology.
Rights and permissions
About this article
Cite this article
Kahle, M. Random Geometric Complexes. Discrete Comput Geom 45, 553–573 (2011). https://doi.org/10.1007/s00454-010-9319-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-010-9319-3