Alpha Shape Topology of the Cosmic Web

Weygaert, Rien van de; Platen, Erwin; Vegter, Gert; Eldering, Bob; Kruithof, Nico

doi:10.1109/isvd.2010.24

Cited by 22 publications

(14 citation statements)

References 27 publications

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“…Most of the previous literatures analyzed the cellular network deployments from the perspective of BS spatial density distribution by using straightforward simulation methods. As one of the unprecedented researches, this paper introduces a powerful algebraic geometric tool, namely α-Shapes [20], into the fundamental analysis of real BS location data for twelve countries around the world, and generalizes essential topological characteristics through the mass data sets from the perspective of topological invariants, i.e., Betti numbers and Euler characteristics [20]. Briefly speaking, α-Shapes are geometric manifolds constructed from a specific point set so that they are closely related to topological nature of the point set.…”

Section: B Our Contributionsmentioning

confidence: 99%

“…In general, unlike common tools in algebraic topology field, which have to resort to some kind of user-defined smoothing functions or thresholds for the analysis of a discrete point set, α-Shapes depend entirely on the point distribution itself, and focus on the features determined by the set of points exclusively [20]. In this regard, α-Shapes are significantly superior and are regarded as the optimal technique for our study for the same reason.…”

Section: A Alpha Shapes (α-Shapes)mentioning

confidence: 99%

“…The basic notion of homology needs to be stated first for a better elaboration of the Betti numbers. Homology describes the connectivity of space via the characterization of two fundamental morphological elements, namely, holes and boundaries [31]. In terms of three-dimensional space, a 0dimensional hole refers to an independent component or a gap between two disconnected components; a 1-dimensional hole means a tunnel which appears when an edge is added to two connected points, and the tunnel can be crossed over in either direction without running into a boundary; a 2-dimensional hole is a cavity or void, which is completely encircled by a 2-dimensional surface.…”

Section: B Betti Numbersmentioning

confidence: 99%

See 2 more Smart Citations

Study on Base Station Topology in National Cellular Networks: Take Advantage of Alpha Shapes, Betti Numbers, and Euler Characteristics

Chen

Zhao

et al. 2020

IEEE Systems Journal

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Faced with the ever-increasing trend of the cellular network scale, how to quantitatively evaluate the effectiveness of the large-scale deployment of base stations (BSs) has become a challenging topic. To this end, a deeper understanding of the cellular network topology is of fundamental significance to be achieved. In this paper, α-Shape, a powerful algebraic geometric tool, is integrated into the analysis of real BS location data for six Asian countries and six European countries, respectively. Firstly, the BS spatial deployments of both Asian and European countries express fractal features based on two different testifying metrics, namely the Betti numbers and the Hurst coefficients. Secondly, it is found out that the log-normal distribution presents the best match to the cellular network topology when the practical BS deployment is characterized by the Euler characteristics.

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Section: B Our Contributionsmentioning

confidence: 99%

Section: A Alpha Shapes (α-Shapes)mentioning

confidence: 99%

Section: B Betti Numbersmentioning

confidence: 99%

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Study on Base Station Topology in National Cellular Networks: Take Advantage of Alpha Shapes, Betti Numbers, and Euler Characteristics

Chen

Zhao

et al. 2020

IEEE Systems Journal

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“…The Euler characteristics can be calculated from Betti numbers according to Euler-Poincare Formula [6]. Since a clear heavy-tailed property is demonstrated in the probability density functions (PDFs) of the Euler characteristics of the real BS location data, three classical heavy-tailed statistical distributions and widely-used Poisson distribution are selected as the candidates to match the PDFs.…”

Section: Log-normal Distribution Of the Euler Characteristicsmentioning

confidence: 99%

“…However, the majority of related documents studied BS deployments only by means of analyzing BS density distributions. In our works, unprecedentedly, several principal concepts in algebraic geometry field, i.e., α-Shapes, Betti numbers, and Euler characteristics [6], are merged into the analyses of the BS topology of eight representative cities around the world. In essential, BSs can be abstracted into a discrete point set in the 2-dimensional plane, and connections between BSs can be specifically established according to the construction of α-Shapes, so α-Shapes are capable of reflecting topological features of BS deployments.…”

Section: Introductionmentioning

confidence: 99%

Fundamentals on Base Stations in Urban Cellular Networks: From the Perspective of Algebraic Topology

Chen

Zhao

et al. 2019

IEEE Wireless Commun. Lett.

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In recent decades, the deployments of cellular networks have been going through an unprecedented expansion. In this regard, it is beneficial to acquire profound knowledge of cellular networks from the view of topology so that prominent network performances can be achieved by means of appropriate placements of base stations (BSs). In our researches, practical location data of BSs in eight representative cities are processed with classical algebraic geometric instruments, including α-Shapes, Betti numbers, and Euler characteristics. At first, the fractal nature is revealed in the BS topology from both perspectives of the Betti numbers and the Hurst coefficients. Furthermore, lognormal distribution is affirmed to provide the optimal fitness to the Euler characteristics of real BS deployments.

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Alpha, Betti and the Megaparsec Universe: On the Topology of the Cosmic Web

Weygaert

Vegter

Edelsbrunner

et al. 2011

Lecture Notes in Computer Science

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Abstract. We study the topology of the Megaparsec Cosmic Web in terms of the scale-dependent Betti numbers, which formalize the topological information content of the cosmic mass distribution. While the Betti numbers do not fully quantify topology, they extend the information beyond conventional cosmological studies of topology in terms of genus and Euler characteristic. The richer information content of Betti numbers goes along the availability of fast algorithms to compute them. For continuous density fields, we determine the scale-dependence of Betti numbers by invoking the cosmologically familiar filtration of sublevel or superlevel sets defined by density thresholds. For the discrete galaxy distribution, however, the analysis is based on the alpha shapes of the particles. These simplicial complexes constitute an ordered sequence of nested subsets of the Delaunay tessellation, a filtration defined by the scale parameter, α. As they are homotopy equivalent to the sublevel sets of the distance field, they are an excellent tool for assessing the topological structure of a discrete point distribution. In order to develop an intuitive understanding for the behavior of Betti numbers as a function of α, and their relation to the morphological patterns in the Cosmic Web, we first study them within the context of simple heuristic Voronoi clustering models. These can be tuned to consist of specific morphological elements of the Cosmic Web, i.e. clusters, filaments, or sheets. To elucidate the relative prominence of the various Betti numbers in different stages of morphological evolution, we introduce the concept of alpha tracks. Subsequently, we address the topology of structures emerging in the standard LCDM scenario and in cosmological scenarios with alternative dark energy conarXiv:1306.3640v1 [astro-ph.CO] 16 Jun 2013 2 tent. The evolution of the Betti numbers is shown to reflect the hierarchical evolution of the Cosmic Web. We also demonstrate that the scale-dependence of the Betti numbers yields a promising measure of cosmological parameters, with a potential to help in determining the nature of dark energy and to probe primordial non-Gaussianities. We also discuss the expected Betti numbers as a function of the density threshold for superlevel sets of a Gaussian random field. Finally, we introduce the concept of persistent homology. It measures scale levels of the mass distribution and allows us to separate small from large scale features. Within the context of the hierarchical cosmic structure formation, persistence provides a natural formalism for a multiscale topology study of the Cosmic Web.

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Alpha Shape Topology of the Cosmic Web

Cited by 22 publications

References 27 publications

Study on Base Station Topology in National Cellular Networks: Take Advantage of Alpha Shapes, Betti Numbers, and Euler Characteristics

Study on Base Station Topology in National Cellular Networks: Take Advantage of Alpha Shapes, Betti Numbers, and Euler Characteristics

Fundamentals on Base Stations in Urban Cellular Networks: From the Perspective of Algebraic Topology

Alpha, Betti and the Megaparsec Universe: On the Topology of the Cosmic Web

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