On the vanishing of homology in random Čech complexes

We compute the homology of random \v{C}ech complexes over a homogeneous Poisson process on the d-dimensional torus, and show that there are, coarsely, two phase transitions. The first transition is analogous to the Erd\H{o}s-R\'enyi phase transition, where the \v{C}ech complex becomes connected. The second transition is where all the other homology groups are computed correctly (almost simultaneously). Our calculations also suggest a finer measurement of scales, where there is a further refinement to this picture and separation between different homology groups.

[1]  J. Dall,et al.  Random geometric graphs. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  Jeff Cheeger,et al.  Critical points of distance functions and applications to geometry , 1991 .

[3]  K. Borsuk.  On the imbedding of systems of compacta in simplicial complexes , 1948 .

[4]  N. Wallach,et al.  Homological connectivity of random k-dimensional complexes , 2009 .

[5]  Nathan Linial,et al.  Collapsibility and Vanishing of Top Homology in Random Simplicial Complexes , 2010, Discret. Comput. Geom..

[6]  R. Ho.  Algebraic Topology , 2022 .

[7]  Frédéric Chazal,et al.  A Sampling Theory for Compact Sets in Euclidean Space , 2006, SCG '06.

[8]  Nathan Linial,et al.  Homological Connectivity Of Random 2-Complexes , 2006, Comb..

[9]  H. Edelsbrunner,et al.  Topological data analysis , 2011 .

[10]  Matthew Kahle,et al.  Topology of random geometric complexes: a survey , 2014, J. Appl. Comput. Topol..

[11]  Elizabeth S. Meckes,et al.  Limit theorems for Betti numbers of random simplicial complexes , 2010 .

[12]  N. Linial,et al.  Random simplicial complexes - around the phase transition , 2016, 1609.00914.

[13]  D. Yogeshwaran,et al.  On the topology of random complexes built over stationary point processes. , 2012, 1211.0061.

[14]  Geoffrey E. Hinton,et al.  Distance Functions , 2010, Encyclopedia of Machine Learning.

[15]  R. Meshulam,et al.  Homological connectivity of random k-dimensional complexes , 2009, Random Struct. Algorithms.

[16]  R. Adler,et al.  Persistent homology for random fields and complexes , 2010, 1003.1001.

[17]  B. Ripley,et al.  Introduction to the Theory of Coverage Processes. , 1989 .

[18]  Leopold Flatto,et al.  Random coverings , 1977 .

[19]  Gunnar E. Carlsson,et al.  Topology and data , 2009 .

[20]  L. Wasserman.  Topological Data Analysis , 2016, 1609.08227.

[21]  Matthew Kahle,et al.  Sharp vanishing thresholds for cohomology of random flag complexes , 2012, 1207.0149.

[22]  James R. Munkres,et al.  Elements of algebraic topology , 1984 .

[23]  J. Cheeger,et al.  Geometric Topology: Recent Developments , 1991 .

[24]  Matthew Kahle,et al.  Random Geometric Complexes , 2009, Discret. Comput. Geom..

[25]  R. Adler,et al.  Limit Theorems for Point Processes under Geometric Constraints (and Topological Crackle) , 2015, 1503.08416.

[26]  Nathan Linial,et al.  On the phase transition in random simplicial complexes , 2014, 1410.1281.

[27]  Matthew Kahle,et al.  The fundamental group of random 2-complexes , 2007, 0711.2704.

[28]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[29]  Boris G. Pittel,et al.  Inside the critical window for cohomology of random k‐complexes , 2013, Random Struct. Algorithms.

[30]  S. Li.  Concise Formulas for the Area and Volume of a Hyperspherical Cap , 2011 .

[31]  Daniel C. Cohen,et al.  Topology of random 2-complexes , 2010, 1006.4229.

[32]  Sivaraman Balakrishnan,et al.  Tight Lower Bounds for Homology Inference , 2013, ArXiv.

[33]  R. Ghrist.  Barcodes: The persistent topology of data , 2007 .

[34]  R. Adler,et al.  PR ] 2 4 Ju l 2 01 1 Submitted to the Annals of Applied Probability DISTANCE FUNCTIONS , CRITICAL POINTS , AND TOPOLOGY FOR SOME RANDOM COMPLEXES By , 2011 .

[35]  Stephen Smale,et al.  A Topological View of Unsupervised Learning from Noisy Data , 2011, SIAM J. Comput..

[36]  Nathan Linial,et al.  When does the top homology of a random simplicial complex vanish? , 2012, Random Struct. Algorithms.

[37]  Hugues Randriam,et al.  Simplicial Homology of Random Configurations , 2011, Advances in Applied Probability.

[38]  Sivaraman Balakrishnan,et al.  Minimax rates for homology inference , 2011, AISTATS.

[39]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[40]  J. Møller,et al.  Lectures on Random Voronoi Tessellations , 1994 .

[41]  Jesse Freeman,et al.  in Morse theory, , 1999 .

[42]  R. Adler,et al.  Random geometric complexes in the thermodynamic regime , 2014, Probability Theory and Related Fields.

[43]  J. Seaman.  Introduction to the theory of coverage processes , 1990 .

[44]  D. Kozlov.  The threshold function for vanishing of the top homology group of random $d$-complexes , 2009, 0904.1652.

[45]  Omer Bobrowski,et al.  Crackle: The Homology of Noise , 2014, Discret. Comput. Geom..

[46]  S. Mukherjee,et al.  The topology of probability distributions on manifolds , 2013, 1307.1123.

[47]  Stephen Smale,et al.  Finding the Homology of Submanifolds with High Confidence from Random Samples , 2008, Discret. Comput. Geom..

[48]  Michael Farber,et al.  The asphericity of random 2‐dimensional complexes , 2012, Random Struct. Algorithms.

[49]  MORSE THEORY FOR MIN-TYPE FUNCTIONS* , 2016 .