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Simplicial Homology of Random Configurations

Part of: Stochastic analysis Stochastic processes Applied homological algebra and category theory

Published online by Cambridge University Press:  22 February 2016

L. Decreusefond ,
E. Ferraz ,
H. Randriambololona  and
A. Vergne
L. Decreusefond*
Affiliation:
Telecom ParisTech
E. Ferraz*
Affiliation:
Rouen University
H. Randriambololona*
Affiliation:
Telecom ParisTech
A. Vergne*
Affiliation:
Telecom ParisTech
*
∗ Postal address: Institut Mines-Telecom, Telecom ParisTech, CNRS LTCI, 46 rue Barrault, Paris, 75634, France.
∗∗∗ Postal address: Rouen University, LMRS, CNRS UMR 6085, Avenue de l'université, BP 12, Saint-Etienne du Rouvray, 76801, France. Email address: [email protected]
∗ Postal address: Institut Mines-Telecom, Telecom ParisTech, CNRS LTCI, 46 rue Barrault, Paris, 75634, France.
∗ Postal address: Institut Mines-Telecom, Telecom ParisTech, CNRS LTCI, 46 rue Barrault, Paris, 75634, France.
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Abstract

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Given a Poisson process on a d-dimensional torus, its random geometric simplicial complex is the complex whose vertices are the points of the Poisson process and simplices are given by the C̆ech complex associated to the coverage of each point. By means of Malliavin calculus, we compute explicitly the three first-order moments of the number of k-simplices, and provide a way to compute higher-order moments. Then we derive the mean and the variance of the Euler characteristic. Using the Stein method, we estimate the speed of convergence of the number of occurrences of any connected subcomplex as it converges towards the Gaussian law when the intensity of the Poisson point process tends to infinity. We use a concentration inequality for Poisson processes to find bounds for the tail distribution of the Betti number of first order and the Euler characteristic in such simplicial complexes.

Keywords

C̆ech complexconcentration inequalityhomologyMalliavin calculuspoint processRips-Vietoris complex

MSC classification

Primary: 60G55: Point processes
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus 55U10: Simplicial sets and complexes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 46 , Issue 2 , June 2014 , pp. 325 - 347
Copyright
© Applied Probability Trust 

Footnotes

Partially supported by ANR Masterie.

References

Armstrong, M. A. (1979). Basic Topology. McGraw-Hill, London.Google Scholar
Bell, E. T. (1934). Exponential polynomials. Ann. Math. 35, 258–277.CrossRefGoogle Scholar
Bhattacharya, R. N. and Ghosh, J. K. (1992). A class of U-statistics and asymptotic normality of the number of k-clusters. J. Multivariate Anal. 43, 300–330.Google Scholar
Björner, A. (1995). Topological methods. In Handbook of Combinatorics, Elsevier, Amsterdam, pp. 1819–1872.Google Scholar
Cohen-Steiner, D., Edelsbrunner, H. and Harer, J. (2007). Stability of persistence diagrams. Discrete Comput. Geom. 37, 103–120.CrossRefGoogle Scholar
Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
De Silva, V. and Ghrist, R. (2006). Coordinate-free coverage in sensor networks with controlled boundaries via homology. Internat. J. Robotics Res. 25, 1205–1222.CrossRefGoogle Scholar
Decreusefond, L. and Ferraz, E. (2011). On the one dimensional Poisson random geometric graph. J. Prob. Statist. 2011, 350–382.Google Scholar
Decreusefond, L., Joulin, A. and Savy, N. (2010). Upper bounds on Rubinstein distances on configuration spaces and applications. Commun. Stoch. Anal. 4, 377–399.Google Scholar
Edelsbrunner, H., Letscher, D. and Zomorodian, A. (2002). Topological persistence and simplification. Discrete Comput. Geom. 28, 511–533.Google Scholar
Ghrist, R. (2005). Coverage and hole-detection in sensor networks via homology. In Fouth Internat. Conf. Inf. Process. Sensor Networks (IPSN'05), UCLA, pp. 254–260.Google Scholar
Greenberg, M. J. and Harper, J. R. (1981). Algebraic Topology. A First Course. Benjamin/Cummings, Reading, MA.Google Scholar
Hatcher, A. (2002). Algebraic Topology. Cambridge University Press.Google Scholar
Ito, Y. (1988). Generalized Poisson functionals. Prob. Theory Relat. Fields 77, 1–28.Google Scholar
Kahle, M. (2011). Random geometric complexes. Discrete Comput. Geom. 45, 553–573.CrossRefGoogle Scholar
Kahle, M. and Meckes, E. (2013). Limit theorems for Betti numbers of random simplicial complexes. Homology Homotopy Appl. 15, 343–374.Google Scholar
Kahn, J. M., Katz, R. H. and Pister, K. S. J. (1999). Next century challenges: mobile networking for smart dust. In Internat. Conf. Mobile Comput. Networking, Seattle, WA, pp. 271–278.Google Scholar
Lewis, F. (2004). Wireless Sensor Networks. John Wiley, New York.Google Scholar
Munkres, J. R. (1984). Elements of Algebraic Topology. Addison Wesley, Menlo Park, CA.Google Scholar
Nourdin, I. and Peccati, G. (2012). Normal Approximations with Malliavin Calculus. From Stein's Method to Universality. Cambridge University Press.Google Scholar
Peccati, G., Solé, J. L., Taqqu, M. S. and Utzet, F. (2010). Stein's method and normal approximation of Poisson functionals. Ann. Prob. 38, 443–478.Google Scholar
Penrose, M. (2003). Random Geometric Graphs (Oxford Stud. Prob. 5). Oxford University Press.Google Scholar
Penrose, M. D. and Yukich, J. E. (2013). Limit theory for point processes in manifolds. Ann. Appl. Prob. 23, 2161–2211.Google Scholar
Pottie, G. J. and Kaiser, W. J. (2000). Wireless integrated network sensors. Commun. ACM 43, 51–58.Google Scholar
Privault, N. (2009). Stochastic Analysis in Discrete and Continuous Settings with Normal Martingales (Lecture Notes Math. 1982). Springer, Berlin.Google Scholar
Reitzner, M. and Schulte, M. (2013). Central limit theorems for U-statistics of Poisson point processes. Ann. Prob. 41, 3879–3909.CrossRefGoogle Scholar
Rotman, J. (1988). An Introduction to Algebraic Topology (Graduate Texts Math. 119). Springer, New York.Google Scholar