Topology of random geometric complexes: a survey

Bobrowski, Omer; Kahle, Matthew

doi:10.1007/s41468-017-0010-0

Cited by 108 publications

(95 citation statements)

References 49 publications

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“…Interestingly, geometric graphs -which can be used to instantiate spatial constraints on the topology -show a markedly different distribution. There are many low dimensional cavities, and fewer cavities with increasing dimension [43,44,121,123] ( Fig. Box 2, panel C).…”

Section: Box 2: Applied Algebraic Topologymentioning

confidence: 99%

Spatial Embedding Imposes Constraints on Neuronal Network Architectures

Stiso

Bassett

2018

Trends in Cognitive Sciences

106

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Section: Box 2: Applied Algebraic Topologymentioning

confidence: 99%

Spatial Embedding Imposes Constraints on Neuronal Network Architectures

Stiso

Bassett

2018

Trends in Cognitive Sciences

106

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“…The related numerical analysis of the persistent homology of the set of Gaussian field realizations presented in this paper is the subject of the upcoming related article . Our work follows up on early explorations of Gaussian field homology by Adler & Bobrowski (Adler et al 2010;Bobrowski 2012;Bobrowski & Borman 2012). These studies address fundamental and generic aspects and are strongly analytically inclined, but also give numerical results on Gaussian field homology.…”

Section: Introductionmentioning

confidence: 97%

Topology and geometry of Gaussian random fields I: on Betti numbers, Euler characteristic, and Minkowski functionals

Pranav

Weygaert

Vegter

et al. 2019

Monthly Notices of the Royal Astronomical Society

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This study presents a numerical analysis of the topology of a set of cosmologically interesting three-dimensional Gaussian random fields in terms of their Betti numbers β 0 , β 1 and β 2 . We show that Betti numbers entail a considerably richer characterization of the topology of the primordial density field. Of particular interest is that Betti numbers specify which topological features -islands, cavities or tunnels -define its spatial structure.A principal characteristic of Gaussian fields is that the three Betti numbers dominate the topology at different density ranges. At extreme density levels, the topology is dominated by a single class of features. At low levels this is a Swiss-cheeselike topology, dominated by isolated cavities, at high levels a predominantly Meatball-like topology of isolated objects. At moderate density levels, two Betti number define a more Sponge-like topology. At mean density, the topology even needs three Betti numbers, quantifying a field consisting of several disconnected complexes, not of one connected and percolating overdensity.A second important aspect of Betti number statistics is that they are sensitive to the power spectrum. It reveals a monotonic trend in which at a moderate density range a lower spectral index corresponds to a considerably higher (relative) population of cavities and islands.We also assess the level of complementary information that Betti numbers represent, in addition to conventional measures such as Minkowski functionals. To this end, we include an extensive description of the Gaussian Kinematic Formula (GKF), which represents a major theoretical underpinning for this discussion.

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“…The emerging research area known as random topology comprises theoretical results that characterize the asymptotic behavior of topological properties of random objects [2,3,12,13,25,26]. In addition to the mathematical value, such results also find many applications in manifold learning and topological data analysis as they provide tools for interpreting complex high dimensional data sets (see e.g.…”

Section: Introductionmentioning

confidence: 99%

Strong Law of Large Numbers for Betti Numbers in the Thermodynamic Regime

Goel¹,

Trinh²,

Tsunoda³

2018

J Stat Phys

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We establish the strong law of large numbers for Betti numbers of random Cech complexes built on R N -valued binomial point processes and related Poisson point processes in the thermodynamic regime. Here we consider both the case where the underlying distribution of the point processes is absolutely continuous with respect to the Lebesgue measure on R N and the case where it is supported on a C 1 compact manifold of dimension strictly less than N . The strong law is proved under very mild assumption which only requires that the common probability density function belongs to L p spaces, for all 1 ≤ p < ∞.

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Topology of random geometric complexes: a survey

Cited by 108 publications

References 49 publications

Spatial Embedding Imposes Constraints on Neuronal Network Architectures

Spatial Embedding Imposes Constraints on Neuronal Network Architectures

Topology and geometry of Gaussian random fields I: on Betti numbers, Euler characteristic, and Minkowski functionals

Strong Law of Large Numbers for Betti Numbers in the Thermodynamic Regime

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