3D Kinetic Alpha Complexes and Their Implementation

Kerber, Michael; Edelsbrunner, Herbert

doi:10.1137/1.9781611972931.6

Cited by 3 publications

(5 citation statements)

References 13 publications

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“…Given a simplex σ, which can be a point, an edge, a triangle or a tetrahedron, denote the open ball bounded by its minimal circumsphere as B σ . The simplex σ is called Gabriel ( [24]) if B σ ∩ P = ∅. Note that for vertices (0-simplices) the circumradius is considered 0.…”

Section: Complexmentioning

confidence: 99%

Persistent spectral graph

Wang

Nguyen

Wei

2020

Numer Methods Biomed Eng

106

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Persistent homology is constrained to purely topological persistence, while multiscale graphs account only for geometric information. This work introduces persistent spectral theory to create a unified low‐dimensional multiscale paradigm for revealing topological persistence and extracting geometric shapes from high‐dimensional datasets. For a point‐cloud dataset, a filtration procedure is used to generate a sequence of chain complexes and associated families of simplicial complexes and chains, from which we construct persistent combinatorial Laplacian matrices. We show that a full set of topological persistence can be completely recovered from the harmonic persistent spectra, that is, the spectra that have zero eigenvalues, of the persistent combinatorial Laplacian matrices. However, non‐harmonic spectra of the Laplacian matrices induced by the filtration offer another powerful tool for data analysis, modeling, and prediction. In this work, fullerene stability is predicted by using both harmonic spectra and non‐harmonic persistent spectra, while the latter spectra are successfully devised to analyze the structure of fullerenes and model protein flexibility, which cannot be straightforwardly extracted from the current persistent homology. The proposed method is found to provide excellent predictions of the protein B‐factors for which current popular biophysical models break down.

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Section: Complexmentioning

confidence: 99%

Persistent spectral graph

Wang

Nguyen

Wei

2020

Numer Methods Biomed Eng

106

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“…In Figure 2 (2,14,12,17,19). The other three boundary cycles in this example are (4,20,21,22,18,10,15), (6,16,8), and (1, 3, 5, 7, 9, 11, 13); see Figure 2 (right).…”

Section: Fat Graphs and Boundary Cyclesmentioning

confidence: 86%

“…In our example, we compute that ϕ(2) = 14, ϕ(14) = 12, ϕ(12) = 17, ϕ(17) = 19, and ϕ(19) = 2, so one of the boundary cycles is the orbit(2,14,12,17,19). The other three boundary cycles in this example are(4,20,21,22,18,10,15),(6,16,8), and (1,3,5,7,9,11,13); see Figure2 (right).We note that each boundary cycle corresponds to a connected component of the complement of the graph. Therefore, the boundary cycles will allow us to identify the Reeb graph of the uncovered region in spacetime.…”

mentioning

confidence: 83%

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Efficient Evader Detection in Mobile Sensor Networks

Adams,

Ghosh,

Mask

et al. 2021

Preprint

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Suppose one wants to monitor a domain with sensors each sensing small ball-shaped regions, but the domain is hazardous enough that one cannot control the placement of the sensors. A prohibitively large number of randomly placed sensors would be required to obtain static coverage. Instead, one can use fewer sensors by providing only mobile coverage, a generalization of the static setup wherein every possible intruder is detected by the moving sensors in a bounded amount of time. Here, we use topology in order to implement algorithms certifying mobile coverage that use only local data to solve the global problem. Our algorithms do not require knowledge of the sensors' locations. We experimentally study the statistics of mobile coverage in two dynamical scenarios. We allow the sensors to move independently (billiard dynamics and Brownian motion), or to locally coordinate their dynamics (collective animal motion models). Our detailed simulations show, for example, that collective motion enhances performance: The expected time until the mobile sensor network achieves mobile coverage is lower for the D'Orsogna collective motion model than for the billiard motion model. Further, we show that even when the probability of static coverage is low, all possible evaders can nevertheless be detected relatively quickly by mobile sensors.

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“…For DC s, deciding when a k-simplex T , k ≤ 3, becemes (seizes to be) Delaunay involves finding roots of polynomials of 8-th degree (Russel, 2007). In α-complexes, it is necessary to determine when a k-simplex T , k ≤ 3, becomes (seizes to be) Gabriel and when it becomes (seizes to be) short (Kerber and Edelsbrunner, 2013).…”

Section: Discussionmentioning

confidence: 99%

Alpha Complexes in Protein Structure Prediction

Winter

Fonseca

2015

Proceedings of the International Conference on Bioinformatics Models, Methods and Algorithms

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Reducing the computational effort and increasing the accuracy of potential energy functions is of utmost importance in modeling biological systems, for instance in protein structure prediction, docking or design. Evaluating interactions between nonbonded atoms is the bottleneck of such computations. It is shown that local properties of α-complexes (subcomplexes of Delaunay tessellations) make it possible to identify nonbonded pairs of atoms whose contributions to the potential energy are not marginal and cannot be disregarded. Computational experiments indicate that using the local properties of α-complexes, the relative error (when compared to the potential energy contributions of all nonbonded pairs of atom) is well within 2%. Furthermore, the computational effort (assuming that α-complexes are given) is comparable to even the simplest and therefore also fastest cutoff approaches. The determination of α-complexes from scratch for every configuration encountered during the search for the native structure would make this approach hopelessly slow. However, it is argued that kinetic α-complexes can be used to reduce the computational effort of determining the potential energy when "moving" from one configuration to a neighboring one. As a consequence, relatively expensive (initial) construction of an α-complex is expected to be compensated by subsequent fast kinetic updates during the search process. Computational results presented in this paper are limited. However, they suggest that the applicability of αcomplexes and kinetic α-complexes in protein related problems (e.g., protein structure prediction and proteinligand docking) deserves furhter investigation.

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3D Kinetic Alpha Complexes and Their Implementation

Cited by 3 publications

References 13 publications

Persistent spectral graph

Persistent spectral graph

Efficient Evader Detection in Mobile Sensor Networks

Alpha Complexes in Protein Structure Prediction

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