Random Geometric Complexes

Kahle, Matthew

doi:10.1007/s00454-010-9319-3

Cited by 153 publications

(210 citation statements)

References 28 publications

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“…We conclude this section by noting that the simulation results also support an open conjectures about the Betti curves being unimodal [22]. If the above holds over a large enough set of parameters, the shape of the Euler curve may be used to show unimodality of the Betti curves.…”

Section: Resultssupporting

confidence: 64%

Homological percolation and the Euler characteristic

Bobrowski

Škraba

2020

Phys. Rev. E

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In this paper we study the connection between the phenomenon of homological percolation (the formation of "giant" cycles in persistent homology), and the zeros of the expected Euler characteristic curve. We perform an experimental study that covers four different models: site-percolation on the cubical and permutahedral lattices, the Poisson-Boolean model, and Gaussian random fields.All the models are generated on the flat torus T d , for d = 2, 3, 4. The simulation results strongly indicate that the zeros of the expected Euler characteristic curve approximate the critical values for homological-percolation. Our results also provide some insight about the approximation error.Further study of this connection could have powerful implications both in the study of percolation theory, and in the field of Topological Data Analysis.

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Section: Resultssupporting

confidence: 64%

Homological percolation and the Euler characteristic

Bobrowski

Škraba

2020

Phys. Rev. E

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“…To place this paper in a more general context, we mention that there has recently been quite a bit of exciting work on topologically constructed probability spaces; we refer in particular to [6,7,8,9,12,14] and the references therein.…”

Section: Thresholds For Vanishing Of the (D − 1)st Homology Group Of mentioning

confidence: 99%

The threshold function for vanishing of the top homology group of random 𝑑-complexes

Kozlov

2010

Proc. Amer. Math. Soc.

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Abstract. For positive integers n and d, and the probability function 0 ≤ p(n) ≤ 1, we let Y n,p,d denote the probability space of all at most d-dimensional simplicial complexes on n vertices, which contain the full (d − 1)-dimensional skeleton, and whose d-simplices appear with probability p(n). In this paper we determine the threshold function for vanishing of the top homology group in Y n,p,d , for all d ≥ 1. Thresholds for vanishing of the (d − 1)st homology group of random d-complexesIn 1959 Erdős and Rényi defined a natural model for random graphs which has since become classical. In this model, which we call 1 Y n,p,1 , the random graph has n vertices, and the edges are chosen uniformly and independently at random with probability p. Usually, one is interested in questions concerning various statistics on this probability space, in the situation when n goes to infinity, and p is a function of n. One of the main results of Erdős-Rényi concerning Y n,p,1 was the discovery of the threshold function for the connectivity of the graph. More precisely, reformulated in our language, they have shown the following theorem. Theorem 1.1 (Erdős-Rényi Theorem, [4]). Assume that w(n) is any functionw : N → R, such that lim n→∞ w(n) = ∞, and p = p(n) is the probability depending on n. Then we haveMore recently, the two-dimensional analog Y n,p,2 of the Erdős-Rényi model was considered by Linial-Meshulam in [11], and, further, the d-dimensional model Y n,p,d , for d ≥ 3, was considered by Meshulam-Wallach in [13].

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“…We will refer to these generalizations as random combinatorial complexes (see [26] for a survey). It turns out that the Erdős -Rényi threshold for connectivity can be generalized to that of "homological connectivity," where the higher homology groups H k become trivial.In parallel to the study of combinatorial complexes, a line of research was established for random geometric complexes [3,5,24,42,43]. This type of complexes generalizes the model of the random geometric graph G(n, r) (introduced by Gilbert [18]), where vertices are placed at random in a metric-measure space, and edges are included based on proximity [38].…”

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confidence: 99%

Random Čech complexes on Riemannian manifolds

Bobrowski

Oliveira

2018

Random Struct Algorithms

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In this paper we study the homology of a random Čech complex generated by a homogeneous Poisson process in a compact Riemannian manifold M. In particular, we focus on the phase transition for “homological connectivity” where the homology of the complex becomes isomorphic to that of M. The results presented in this paper are an important generalization of , from the flat torus to general compact Riemannian manifolds. In addition to proving the statements related to homological connectivity, the methods we develop in this paper can be used as a framework for translating results for random geometric graphs and complexes from the Euclidean setting into the more general Riemannian one.

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Random Geometric Complexes

Cited by 153 publications

References 28 publications

Homological percolation and the Euler characteristic

Homological percolation and the Euler characteristic

The threshold function for vanishing of the top homology group of random 𝑑-complexes

Random Čech complexes on Riemannian manifolds

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