Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Thresholds for vanishing of ‘Isolated’ faces in random Čech and Vietoris–Rips complexes

Srikanth K. Iyer and D. Yogeshwaran

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We study combinatorial connectivity for two models of random geometric complexes. These two models – Čech and Vietoris–Rips complexes – are built on a homogeneous Poisson point process of intensity $n$ on a $d$-dimensional torus, $d>1$, using balls of radius $r_{n}$. In the former, the $k$-simplices/faces are formed by subsets of $(k+1)$ Poisson points such that the balls of radius $r_{n}$ centred at these points have a mutual interesection and in the latter, we require only a pairwise intersection of the balls. Given a (simplicial) complex (i.e., a collection of $k$-simplices for all $k\geq 1$), we can connect $k$-simplices via $(k+1)$-simplices (‘up-connectivity’) or via $(k-1)$-simplices (‘down-connectivity). Our interest is to understand these two combinatorial notions of connectivity for the random Čech and Vietoris–Rips complexes asymptotically as $n\to \infty $. In particular, we analyse in detail the threshold radius for vanishing of isolated $k$-faces for up and down connectivity of both types of random geometric complexes. Though it is expected that the threshold radius $r_{n}=\Theta ((\frac{\log n}{n})^{1/d})$ in coarse scale, our results give tighter bounds on the constants in the logarithmic scale as well as shed light on the possible second-order correction factors. Further, they also reveal interesting differences between the phase transition in the Čech and Vietoris–Rips cases. The analysis is interesting due to non-monotonicity of the number of isolated $k$-faces (as a function of the radius) and leads one to consider ‘monotonic’ vanishing of isolated $k$-faces. The latter coincides with the vanishing threshold mentioned above at a coarse scale (i.e., $\log n$ scale) but differs in the $\log \log n$ scale for the Čech complex with $k=1$ in the up-connected case. For the case of up-connectivity in the Vietoris–Rips complex and for $r_{n}$ in the critical window, we also show a Poisson convergence for the number of isolated $k$-faces when $k\leq d$.


Nous étudions la connectivité combinatoire pour deux modèles de complexes géométriques aléatoires. Ces deux modèles – les complexes de $\v{C}$ech et de Vietoris–Rips – sont construits sur la base d’un processus de Poisson homogène d’intensité $n$ sur un tore de dimension $d$, $d>1$, en utilisant des boules de rayon $r_{n}$. Dans le premier, les $k$-simplexes/faces sont formés par les sous-ensembles de $k+1$ points du processus de Poisson tels que l’intersection des boules de rayon $r_{n}$ centrées en ces points est non vide, et dans le second, nous demandons seulement que les intersections deux-à-deux des boules soient non vides. Étant donné un complexe simplicial (c’est-à-dire une collection de $k$-simplexes pour tous $k\geq 1$), nous pouvons connecter les $k$-simplexes via les $(k+1)$-simplexes (connectivité par le haut) ou via les $(k-1)$-simplexes (connectivité par le bas).

Notre objectif est de comprendre ces deux notions combinatoires de connectivité pour les complexes de Čech et Vietoris–Rips asymptotiquement lorsque $n\to \infty$.

En particulier, nous analysons en détail le rayon critique pour la disparition des $k$-faces isolées pour la connectivité par le haut et par le bas dans les deux types de complexes géométriques aléatoires. Bien qu’il soit attendu que le rayon critique soit $r_{n}=\Theta((\frac{\log n}{n})^{1/d})$ dans une échelle grossière, nos résultats donnent des bornes plus fines sur les constantes dans l’échelle logarithmique et suggère les possibles facteurs correctifs de second ordre. De plus, ils révèlent aussi des différences intéressantes entre les transitions de phase entre les cas de Čech et Vietoris–Rips.

L’analyse est intéressante du fait de la non monotonie du nombre de $k$-faces isolées (comme fonction du rayon) ce qui conduit à considérer une version monotone de la disparition des $k$-faces. Cette dernière coïncide avec le seuil de disparition mentionné précédemment à une échelle grossière (c’est-à-dire à une échelle $\log n$) mais diffère à l’échelle $\log \log n$ pour le complexe de Čech avec $k=1$ pour la connectivité par le haut.

Dans le cas de la connectivité par le haut dans le cas du complexe de Vietoris–Rips et pour $r_{n}$ dans la fenêtre critique, nous montrons aussi une convergence vers un processus de Poisson pour le nombre de $k$-faces isolées quand $k\leq d$.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 56, Number 3 (2020), 1869-1897.

Received: 5 March 2018
Revised: 23 August 2019
Accepted: 25 August 2019
First available in Project Euclid: 26 June 2020

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Mathematical Reviews number (MathSciNet)

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 05E45: Combinatorial aspects of simplicial complexes
Secondary: 60B99: None of the above, but in this section 05C80: Random graphs [See also 60B20]

Random geometric complexes Random hypergraphs Connectivity Maximal faces Phase transition Poisson convergence


Iyer, Srikanth K.; Yogeshwaran, D. Thresholds for vanishing of ‘Isolated’ faces in random Čech and Vietoris–Rips complexes. Ann. Inst. H. Poincaré Probab. Statist. 56 (2020), no. 3, 1869--1897. doi:10.1214/19-AIHP1020.

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  • [1] M. J. B. Appel and R. P. Russo. The connectivity of a graph on uniform points on $[0,1]^{d}$. Statist. Probab. Lett. 60 (2002) 351–357.
  • [2] R. Atkin. An algebra for patterns on a complex, I. Int. J. Man-Mach. Stud. 6 (3) (1974) 285–307.
  • [3] R. Atkin. An algebra for patterns on a complex, II. Int. J. Man-Mach. Stud. 8 (5) (1976) 483–498.
  • [4] H. Barcelo, X. Kramer, R. Laubenbacher and C. Weaver. Foundations of a connectivity theory for simplicial complexes. Adv. in Appl. Math. 26 (2) (2001) 97–128.
  • [5] H. Barcelo and R. Laubenbacher. Perspectives on $A$-homotopy theory and its applications. Discrete Math. 298 (1–3) (2005) 39–61.
  • [6] A. Björner. Topological methods. In Handbook of Combinatorics 1819–1872, 2, 1995.
  • [7] B. Blaszczyszyn and D. Yogeshwaran. Clustering and percolation of point processes. Electron. J. Probab. 18 (2013) 72.
  • [8] O. Bobrowski. Homological connectivity in Cech complexes, 2019. Available at arXiv:1906.04861.
  • [9] O. Bobrowski and R. J. Adler. Distance functions, critical points, and the topology of random Cech complexes. Homology, Homotopy Appl. 16 (2) (2014) 311–344.
  • [10] O. Bobrowski and M. Kahle. Topology of random geometric complexes: A survey. J. Appl. Comput. Top. (2014) 1–34.
  • [11] O. Bobrowski and S. Mukherjee. The topology of probability distributions on manifolds. Probab. Theory Related Fields 161 (3) (2015) 651–686.
  • [12] O. Bobrowski and G. Oliveira. Random Cech complexes on Riemannian manifolds. Random Structures Algorithms 54 (3) (2019) 373–412.
  • [13] O. Bobrowski and S. Weinberger. On the vanishing of homology in random Cech complexes. Random Structures Algorithms 51 (1) (2017) 14–51.
  • [14] B. Bollobás and O. Riordan. Clique percolation. Random Structures Algorithms 35 (3) (2009) 294–322.
  • [15] G. Carlsson. Topological pattern recognition for point cloud data. Acta Numer. 23 (2014) 289–368.
  • [16] A. E. Costa, M. Farber and T. Kappeler. Topics of stochastic algebraic topology. In Proceedings of the Workshop on Geometric and Topological Methods in Computer Science (GETCO) 53–70. Electronic Notes in Theoretical Computer Science 283, 2012.
  • [17] H.-L. de Kergorlay, U. Tillmann and O. Vipond. Random Cech complexes on manifolds with boundary, 2019. Available at arXiv:1906.07626.
  • [18] I. Derényi, G. Palla and T. Vicsek. Clique percolation in random networks. Phys. Rev. Lett. 94 (16) (2005).
  • [19] H. Edelsbrunner and J. L. Harer. Computational Topology, an Introduction. American Mathematical Society, Providence, RI, 2010.
  • [20] P. Erdös and A. Rényi. On random graphs. I. Publ. Math. Debrecen 6 (1959) 290–297.
  • [21] L. Flatto and D. J. Newman. Random coverings. Acta Math. 138 (1) (1977) 241–264.
  • [22] E. N. Gilbert. Random plane networks. SIAM J. Appl. Math. 9 (1961) 533–543.
  • [23] L. Goldstein and M. D. Penrose. Normal approximation for coverage models over binomial point processes. Ann. Appl. Probab. 20 (2010) 696–721.
  • [24] A. Gundert and U. Wagner. On Laplacians of random complexes. In Proceedings of the Twenty-Eighth Annual Symposium on Computational Geometry 151–160. ACM, New York, 2012.
  • [25] B. Gupta and S. K. Iyer. Criticality of the exponential rate of decay for the largest nearest-neighbor link in random geometric graphs. Adv. in Appl. Probab. 42 (3) (2010) 631–658.
  • [26] A. Gut. An Intermediate Course in Probability. Springer, Berlin, 2009.
  • [27] P. Hall. Introduction to the Theory of Coverage Processes. Wiley, New York, 1988.
  • [28] Y. Hiraoka and T. Shirai. Minimum spanning acycle and lifetime of persistent homology in the Linial–Meshulam process. Random Structures Algorithms 51 (2) (2017) 315–340.
  • [29] D. Horak and J. Horst. Spectra of combinatorial Laplace operators on simplicial complexes. Adv. Math. 244 (2013) 303–336.
  • [30] S. K. Iyer and D. Thacker. Nonuniform random geometric graphs with location-dependent radii. Ann. Appl. Probab. 22 (5) (2012) 2048–2066.
  • [31] M. Kahle. Topology of random clique complexes. Discrete Math. 309 (6) (2009) 1658–1671.
  • [32] M. Kahle. Random geometric complexes. Discrete Comput. Geom. 45 (3) (2011) 553–573.
  • [33] M. Kahle. Sharp vanishing thresholds for cohomology of random flag complexes. Ann. of Math. 179 (2014) 1085–1107.
  • [34] M. Kahle. Topology of random simplicial complexes: A survey. Contemp. Math.– Am. Math. Soc. 620 (2014) 201–222.
  • [35] M. Kahle and B. Pittel. Inside the critical window for cohomology of random k-complexes Rand. Struct. Alg. 48 (1) (2014) 102–124.
  • [36] M. Kraetzl, R. Laubenbacher and M. E. Gaston. Combinatorial and algebraic approaches to network analysis. DSTO Internal Report, 2001.
  • [37] N. Linial and R. Meshulam. Homological connectivity of random 2-complexes. Combinatorica 26 (4) (2006) 475–487.
  • [38] R. Meshulam and N. Wallach. Homological connectivity of random k-dimensional complexes. Random Structures Algorithms 34 (3) (2009) 408–417.
  • [39] P. A. P. Moran. The random volume of interpenentrating spheres in space. J. Appl. Probab. 10 (1973) 837–846.
  • [40] S. Mukherjee and J. Steenbergen. Random walks on simplicial complexes and harmonics. Random Structures Algorithms 49 (2) (2016) 379–405.
  • [41] J. R. Munkres. Elements of Algebraic Topology. Addison-Wesley, Reading, 1984.
  • [42] P. Niyogi, S. Smale and S. Weinberger. Finding the homology of submanifolds with high confidence from random samples. Discrete Comput. Geom. 39 (1) (2008) 419–441.
  • [43] T. Owada. Limit theorems for the sum of persistence barcodes. Ann. Appl. Probab. 28 (5) (2018) 2814–2854.
  • [44] T. Owada and R. J. Adler. Limit theorems for point processes under geometric constraints (and topological crackle). Ann. Probab. 45 (3) (2017) 2004–2055.
  • [45] G. Palla, D. Ábel, I. Farkas, P. Pollner, I. Derényi and T. Vicsek. k-clique percolation and clustering. In Handbook of Large-Scale Random Networks 369–408. B. Bollobás, R. Kozma and D. Miklós (Eds). Springer, Berlin Heidelberg, 2008.
  • [46] O. Parzanchevski and R. Rosenthal. Simplicial complexes: Spectrum, homology and random walks. Random Structures Algorithms 50 (2) (2017) 225–261.
  • [47] M. D. Penrose. The longest edge of the random minimal spanning tree. Ann. Appl. Probab. 7 (1997) 340–361.
  • [48] M. D. Penrose. Random Geometric Graphs. Oxford University Press, New York, 2003.
  • [49] M. D. Penrose. Inhomogeneous random graphs, isolated vertices, and Poisson approximation. J. Appl. Probab. 55 (1) (2018) 112–136.
  • [50] R. Schneider and W. Weil. Stochastic and Integral Geometry. Springer, Berlin, 2008.
  • [51] P. Skraba, G. Thoppe and D. Yogeshwaran. Randomly weighted $d$-complexes: Minimal spanning acycles and persistence diagrams. Elec. J. Comb. 27(2) (2020).
  • [52] D. Yogeshwaran and R. J. Adler. On the topology of random complexes built over stationary point processes. Ann. Appl. Probab. 25 (6) (2015) 3338–3380.