Advances in Applied Probability

Simplicial homology of random configurations

L. Decreusefond, E. Ferraz, H. Randriambololona, and A. Vergne

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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 Cech 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.

Article information

Adv. in Appl. Probab., Volume 46, Number 2 (2014), 325-347.

First available in Project Euclid: 29 May 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G55: Point processes
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus 55U10: Simplicial sets and complexes

Cech complex concentration inequality homology Malliavin calculus point process Rips-Vietoris complex


Decreusefond, L.; Ferraz, E.; Randriambololona, H.; Vergne, A. Simplicial homology of random configurations. Adv. in Appl. Probab. 46 (2014), no. 2, 325--347. doi:10.1239/aap/1401369697.

Export citation


  • Armstrong, M. A. (1979). Basic Topology. McGraw-Hill, London.
  • Bell, E. T. (1934). Exponential polynomials. Ann. Math. 35, 258–277.
  • 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.
  • Björner, A. (1995). Topological methods. In Handbook of Combinatorics, Elsevier, Amsterdam, pp. 1819–1872.
  • Cohen-Steiner, D., Edelsbrunner, H. and Harer, J. (2007). Stability of persistence diagrams. Discrete Comput. Geom. 37, 103–120.
  • Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.
  • 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.
  • Decreusefond, L. and Ferraz, E. (2011). On the one dimensional Poisson random geometric graph. J. Prob. Statist. 2011, 350–382.
  • Decreusefond, L., Joulin, A. and Savy, N. (2010). Upper bounds on Rubinstein distances on configuration spaces and applications. Commun. Stoch. Anal. 4, 377–399.
  • Edelsbrunner, H., Letscher, D. and Zomorodian, A. (2002). Topological persistence and simplification. Discrete Comput. Geom. 28, 511–533.
  • 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.
  • Greenberg, M. J. and Harper, J. R. (1981). Algebraic Topology. A First Course. Benjamin/Cummings, Reading, MA.
  • Hatcher, A. (2002). Algebraic Topology. Cambridge University Press.
  • Ito, Y. (1988). Generalized Poisson functionals. Prob. Theory Relat. Fields 77, 1–28.
  • Kahle, M. (2011). Random geometric complexes. Discrete Comput. Geom. 45, 553–573.
  • Kahle, M. and Meckes, E. (2013). Limit theorems for Betti numbers of random simplicial complexes. Homology Homotopy Appl. 15, 343–374.
  • 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.
  • Lewis, F. (2004). Wireless Sensor Networks. John Wiley, New York.
  • Munkres, J. R. (1984). Elements of Algebraic Topology. Addison Wesley, Menlo Park, CA.
  • Nourdin, I. and Peccati, G. (2012). Normal Approximations with Malliavin Calculus. From Stein's Method to Universality. Cambridge University Press.
  • 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.
  • Penrose, M. (2003). Random Geometric Graphs (Oxford Stud. Prob. 5). Oxford University Press.
  • Penrose, M. D. and Yukich, J. E. (2013). Limit theory for point processes in manifolds. Ann. Appl. Prob. 23, 2161–2211.
  • Pottie, G. J. and Kaiser, W. J. (2000). Wireless integrated network sensors. Commun. ACM 43, 51–58.
  • Privault, N. (2009). Stochastic Analysis in Discrete and Continuous Settings with Normal Martingales (Lecture Notes Math. 1982). Springer, Berlin.
  • Reitzner, M. and Schulte, M. (2013). Central limit theorems for $U$-statistics of Poisson point processes. Ann. Prob. 41, 3879–3909.
  • Rotman, J. (1988). An Introduction to Algebraic Topology (Graduate Texts Math. 119). Springer, New York.