The Annals of Applied Probability

On the topology of random complexes built over stationary point processes

Abstract

There has been considerable recent interest, primarily motivated by problems in applied algebraic topology, in the homology of random simplicial complexes. We consider the scenario in which the vertices of the simplices are the points of a random point process in $\mathbb{R}^{d}$, and the edges and faces are determined according to some deterministic rule, typically leading to Čech and Vietoris–Rips complexes. In particular, we obtain results about homology, as measured via the growth of Betti numbers, when the vertices are the points of a general stationary point process. This significantly extends earlier results in which the points were either i.i.d. observations or the points of a Poisson process. In dealing with general point processes, in which the points exhibit dependence such as attraction or repulsion, we find phenomena quantitatively different from those observed in the i.i.d. and Poisson cases. From the point of view of topological data analysis, our results seriously impact considerations of model (non)robustness for statistical inference. Our proofs rely on analysis of subgraph and component counts of stationary point processes, which are of independent interest in stochastic geometry.

Article information

Source
Ann. Appl. Probab., Volume 25, Number 6 (2015), 3338-3380.

Dates
Revised: September 2014
First available in Project Euclid: 1 October 2015

https://projecteuclid.org/euclid.aoap/1443703776

Digital Object Identifier
doi:10.1214/14-AAP1075

Mathematical Reviews number (MathSciNet)
MR3404638

Zentralblatt MATH identifier
1328.60123

Citation

Yogeshwaran, D.; Adler, Robert J. On the topology of random complexes built over stationary point processes. Ann. Appl. Probab. 25 (2015), no. 6, 3338--3380. doi:10.1214/14-AAP1075. https://projecteuclid.org/euclid.aoap/1443703776

References

• [1] Babson, E., Hoffman, C. and Kahle, M. (2011). The fundamental group of random 2-complexes. J. Amer. Math. Soc. 24 1–28.
• [2] Björner, A. (1995). Topological methods. In Handbook of Combinatorics, Vol. 1, 2 1819–1872. Elsevier, Amsterdam.
• [3] Błaszczyszyn, B. and Yogeshwaran, D. (2015). Clustering, percolation and comparison of point processes. In Stochastic Geometry, Spatial Statistics and Random Fields: Models and Algorithms (V. Schmidt, ed.). Lecture Notes in Mathematics 2120. Springer, Cham. To appear.
• [4] Błaszczyszyn, B. and Yogeshwaran, D. (2013). Clustering and percolation of point processes. Electron. J. Probab. 18 no. 72, 20.
• [5] Błaszczyszyn, B. and Yogeshwaran, D. (2014). On comparison of clustering properties of point processes. Adv. in Appl. Probab. 46 1–20.
• [6] Bobrowski, O. and Adler, R. J. (2014). Distance functions, critical points, and topology for some random complexes. Homology, Homotopy Appl. 16 311–344.
• [7] Bobrowski, O. and Mukherjee, S. (2015). The topology of probability distributions on manifolds. Probab. Theory Related Fields 161 651–686.
• [8] Bobrowski, O., Mukherjee, S. and Taylor, J. (2014). Topological consistency via kernel estimation. Preprint. Available at arXiv:1407.5272.
• [9] Bubenik, P., Carlsson, G., Kim, P. T. and Luo, Z.-M. (2010). Statistical topology via Morse theory persistence and nonparametric estimation. In Algebraic Methods in Statistics and Probability II. Contemp. Math. 516 75–92. Amer. Math. Soc., Providence, RI.
• [10] Burton, R. and Waymire, E. (1985). Scaling limits for associated random measures. Ann. Probab. 13 1267–1278.
• [11] Carlsson, G. (2009). Topology and data. Bull. Amer. Math. Soc. (N.S.) 46 255–308.
• [12] Carlsson, G. (2014). Topological pattern recognition for point cloud data. Acta Numer. 23 289–368.
• [13] Chazal, F., Cohen-Steiner, D. and Mérigot, Q. (2011). Geometric inference for probability measures. Found. Comput. Math. 11 733–751.
• [14] Cohen, D., Costa, A., Farber, M. and Kappeler, T. (2011). Topology of random 2-complexes. Discrete Comput. Geom. 47 No. 1.
• [15] Edelsbrunner, H. and Harer, J. L. (2010). Computational Topology: An Introduction. Amer. Math. Soc., Providence, RI.
• [16] Forman, R. (2002). A user’s guide to discrete Morse theory. Sém. Lothar. Combin. 48 Art. B48c, 35.
• [17] Gershkovich, V. and Rubinstein, H. (1997). Morse theory for Min-type functions. Asian J. Math. 1 696–715.
• [18] Ghosh, S. (2012). Determinantal processes and completeness of random exponentials: The critical case. Preprint. Available at arXiv:1211.2435.
• [19] Ghrist, R. (2008). Barcodes: The persistent topology of data. Bull. Amer. Math. Soc. (N.S.) 45 61–75.
• [20] Hough, J. B., Krishnapur, M., Peres, Y. and Virág, B. (2009). Zeros of Gaussian Analytic Functions and Determinantal Point Processes. University Lecture Series 51. Amer. Math. Soc., Providence, RI.
• [21] Kahle, M. (2009). Topology of random clique complexes. Discrete Math. 309 1658–1671.
• [22] Kahle, M. (2011). Random geometric complexes. Discrete Comput. Geom. 45 553–573.
• [23] Kahle, M. (2014). Topology of random simplicial complexes: A survey. In Algebraic Topology: Applications and New Directions. Contemp. Math. 620 (U. Tillmann, S. Galatius and D. Sinha, eds.). Amer. Math. Soc., Providence, RI.
• [24] Kahle, M. and Meckes, E. (2013). Limit theorems for Betti numbers of random simplicial complexes. Homology, Homotopy Appl. 15 343–374.
• [25] Kallenberg, O. (1983). Random Measures, 3rd ed. Akademie-Verlag, Berlin.
• [26] Lavancier, F., Møller, J. and Rubak, E. (2012). Determinantal point process models and statistical inference. Preprint. Available at arXiv:1205.4818.
• [27] Linial, N. and Meshulam, R. (2006). Homological connectivity of random 2-complexes. Combinatorica 26 475–487.
• [28] Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge Tracts in Mathematics 119. Cambridge Univ. Press, Cambridge.
• [29] Meshulam, R. and Wallach, N. (2009). Homological connectivity of random $k$-dimensional complexes. Random Structures Algorithms 34 408–417.
• [30] Mileyko, Y., Mukherjee, S. and Harer, J. (2011). Probability measures on the space of persistence diagrams. Inverse Problems 27 124007, 22.
• [31] Miyoshi, N. and Shirai, T. (2013). Cellular networks with $\alpha$-Ginibre configurated base stations. Available at https://www.researchgate.net/publication/260094649.
• [32] Nazarov, F. and Sodin, M. (2012). Correlation functions for random complex zeroes: Strong clustering and local universality. Comm. Math. Phys. 310 75–98.
• [33] Pemantle, R. (2000). Towards a theory of negative dependence. J. Math. Phys. 41 1371–1390.
• [34] Pemantle, R. and Peres, Y. (2014). Concentration of Lipschitz functionals of determinantal and other strong Rayleigh measures. Combin. Probab. Comput. 23 140–160.
• [35] Penrose, M. (2003). Random Geometric Graphs. Oxford Studies in Probability 5. Oxford Univ. Press, Oxford.
• [36] Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.
• [37] Shirai, T. and Takahashi, Y. (2003). Random point fields associated with certain Fredholm determinants. I. Fermion, Poisson and boson point processes. J. Funct. Anal. 205 414–463.
• [38] Stoyan, D., Kendall, W. and Mecke, J. (1995). Stochastic Geometry and Its Applications. Wiley, Chichester.
• [39] Tausz, A. and Vejdemo-Johansson, M. (2011). JavaPlex: A research software package for persistent (co) homology. Available at http://code.google.com/p/javaplex/.
• [40] Turner, K., Mileyko, Y., Mukherjee, S. and Harer, J. (2014). Fréchet means for distributions of persistence diagrams. Discrete Comput. Geom. 52 44–70.
• [41] Yogeshwaran, D., Subag, E. and Adler, R. J. (2014). Random geometric complexes in the thermodynamic regime. Preprint. Available at arXiv:1403.1164.
• [42] Zomorodian, A. J. (2009). Topology for Computing. Cambridge Monographs on Applied and Computational Mathematics 16. Cambridge Univ. Press, Cambridge.