Abstract
In this expository article, we survey the rapidly emerging area of random geometric simplicial complexes. Random geometric complexes may be viewed as higher-dimensional generalizations of random geometric graphs, where vertices are generated by a random point process, and edges are placed based on proximity. Extending the notion of connected components and cycles in graphs, the main object of study has been the homology of these complexes. We review the results known to date about the probabilistic behavior of the homology (and related structures) generated by these random complexes.
Similar content being viewed by others
References
Adler, R.J., Bobrowski, O., Weinberger, S.: Crackle: the homology of noise. Discret. Comput. Geom. 52(4), 680–704 (2014)
Adler, R.J., Taylor, J.E.: Random fields and geometry, Springer Monographs in Mathematics. Springer, New York (2007)
Adler, R.J., Taylor, J.E.: Topological complexity of smooth random functions. In: Lecture Notes in Mathematics, vol. 2019. Springer, Heidelberg (2011)
Aizenman, M., Chayes, J.T., Chayes, L., Fröhlich, J., Russo L.: On a sharp transition from area law to perimeter law in a system of random surfaces. Comm. Math. Phys. 92(1), 19–69 (1983)
Alon, N., Spencer J.H.: The probabilistic method. Wiley-Interscience Series in Discrete Mathematics and Optimization, 3rd ed. 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. Amer. Math. Soc. 24(1), 1–28 (2011)
Balakrishnan, S., Rinaldo, A., Sheehy, D., Singh, A., Wasserman, L.A.: Minimax rates for homology inference. AISTATS 9, 206–207 (2012)
Baryshnikov, Y., Bubenik, P., Kahle, M.: Min-type Morse theory for configuration spaces of hard spheres. Int. Math. Res. Not. IMRN 9, 2577–2592 (2014). (MR 3207377)
Bobrowski, O., Adler, R.J.: Distance functions, critical points, and the topology of random Čech complexes. Homol. Homotopy Appl. 16(2), 311–344 (2014). (en)
Bobrowski, O., Kahle, M., Skraba, P.: Maximally persistent cycles in random geometric complexes. Ann. Appl. Prob. 27(4), 2032–2060 (2017)
Bobrowski, O., Mukherjee, S.: The topology of probability distributions on manifolds. Probab. Theory Relat Fields 161(3–4), 651–686 (2014)
Bobrowski, O., Mukherjee, S., Taylor, J.E.: Topological consistency via kernel estimation. Bernoulli 23(1), 288–328 (2017)
Bobrowski, O., Oliveira, G.: Random Čech complexes on riemannian manifolds (2017) arXiv:1704.07204 (arXiv preprint)
Bobrowski, O., Weinberger, S.: On the vanishing of homology in random Čech complexes. Random Struct. Algorithm 51(1), 14–51 (2017)
Bollobás, B.: Random graphs, 2nd edn. Cambridge studies in advanced mathematics, vol. 73. Cambridge University Press, Cambridge (2001)
Bollobás, B., Riordan, O.: Percolation. Cambridge University Press, New York (2006)
Borsuk, K.: On the imbedding of systems of compacta in simplicial complexes. Fund. Math. 35, 217–234 (1948)
Bryzgalova, L.N.: The maximum functions of a family of functions that depend on parameters. Funktsional. Anal. i Prilozhen. 12(1), 66–67 (1978)
Bubenik, P., Kim, P.T.: A statistical approach to persistent homology. Homol. Homotopy Appl. 9(2), 337–362 (2007)
Carlsson, G.: Topology and data. Bull. Amer. Math. Soc. (N.S.) 46(2), 255–308 (2009)
De Silva, V., Ghrist, R.: Coverage in sensor networks via persistent homology. Algebr. Geom. Topol. 7, 339–358 (2007)
Decreusefond, L., Ferraz, E., Randriambololona, H., Vergne, A.: Simplicial homology of random configurations. Adv. Appl. Probab. 46(2), 325–347 (2014)
Duy, T.K., Hiraoka, Y., Shirai, T.: Limit theorems for persistence diagrams (2016) arXiv:1612.08371 [math]
Edelsbrunner, H., Kirkpatrick, D.G., Seidel, R.: On the shape of a set of points in the plane. IEEE Trans. Inform. Theory 29(4), 551–559 (1983)
Fasy, B.T., Lecci, F., Rinaldo, A., Wasserman, L., Balakrishnan, S., Singh, A.: Confidence sets for persistence diagrams. Ann. Stat. 42(6), 2301–2339 (2014)
Ganesan, G.: Size of the giant component in a random geometric graph. Ann. Inst. Henri Poincaré Probab. Stat. 49(4), 1130–1140 (2013)
Gershkovich, V., Rubinstein, H.: Morse theory for Min-type functions. Asian J. Math. 1(4), 696–715 (1997)
Ghrist, R.: Barcodes: the persistent topology of data. Bull. Amer. Math. Soc. (N.S.) 45(1), 61–75 (2008)
Gilbert, E.N.: Random plane networks. J. Soc. Indust. Appl. Math. 9, 533–543 (1961)
Grimmett, G.: Percolation, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 321. Springer, Berlin (1999)
Grimmett, G.R., Holroyd, A.E.: Plaquettes, spheres, and entanglement. Electron. J. Probab. 15, 1415–1428 (2010)
Hatcher, A.: Algebraic topology. Cambridge University Press, Cambridge (2002)
Janson, S., Łuczak, T., Rucinski, A.: Random graphs. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York (2000)
Kahle, M.: Random geometric complexes. Discrete Comput. Geom. 45(3), 553–573 (2011)
Kahle, M.: Sharp vanishing thresholds for cohomology of random flag complexes. Ann. Math. (2) 179(3), 1085–1107 (2014)
Kahle, M.: Topology of random simplicial complexes: a survey. AMS Contemp. Math 620, 201–222 (2014)
Kahle, M., Meckes, E.: Limit theorems for Betti numbers of random simplicial complexes. Homol. Homotopy Appl. 15(1), 343–374 (2013)
Kahle, M., Meckes, E.: Erratum: Limit theorems for betti numbers of random simplicial complexes (2015). arXiv:1501.03759 (arXiv preprint)
Linial, N., Meshulam, R.: Homological connectivity of random 2-complexes. Combinatorica 26(4), 475–487 (2006)
Matov, V.I.: Topological classification of the germs of functions of the maximum and minimax of families of functions in general position. Uspekhi Mat. Nauk 37(4)(226), 167–168 (1982)
Meester, R., Roy R.: Continuum percolation, Cambridge Tracts in Mathematics, vol. 119. Cambridge University Press, Cambridge (1996)
Meshulam, R., Wallach, N.: Homological connectivity of random \(k\)-dimensional complexes. Random Struct. Algorithms 34(3), 408–417 (2009)
Milnor, J.: Morse theory. In: Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, no. 51. Princeton University Press, Princeton (1963)
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)
Munkres, J.R.: Elements of algebraic topology. Addison-Wesley Publishing Company, Menlo Park (1984)
Niyogi, P., Smale, S., Weinberger, S.: A topological view of unsupervised learning from noisy data. SIAM J. Comput. 40(3), 646–663 (2011)
Owada, T., Adler, R.J.: Limit theorems for point processes under geometric constraints (and topological crackle). Ann. Prob. 45(3), 2004–2055 (2017)
Penrose, M.: Random geometric graphs, Oxford Studies in Probability, vol. 5. Oxford University Press, Oxford (2003)
Robins, V.: Betti number signatures of homogeneous poisson point processes. Phys. Rev. E 74(6), 061107 (2006)
The GUDHI Project: GUDHI user and reference manual. GUDHI Editorial Board (2015)
Yogeshwaran, D., Adler, R.J.: On the topology of random complexes built over stationary point processes. Ann. Appl. Prob. 25(6), 3338–3380 (2015)
Yogeshwaran, D., Subag, E., Adler, R.J.: Random geometric complexes in the thermodynamic regime. Prob. Theory Relat. Fields 167(1–2), 107–142 (2017)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Additional information
Matthew Kahle gratefully acknowledges support from NSF-DMS #1352386.
Rights and permissions
About this article
Cite this article
Bobrowski, O., Kahle, M. Topology of random geometric complexes: a survey. J Appl. and Comput. Topology 1, 331–364 (2018). https://doi.org/10.1007/s41468-017-0010-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s41468-017-0010-0