Simplicial Models and Topological Inference in Biological Systems

Nanda, Vidit; Sazdanovic, Radmila

doi:10.1007/978-3-642-40193-0_6

Cited by 33 publications

(32 citation statements)

References 28 publications

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“…Vietoris-Rips complexes are a fundamental tool in topological data analysis, they allow to build a topological space from higher dimensional data-set embedded in a metric space [15]. The resulting complex is then studied by persistent homology.…”

Section: Discussionmentioning

confidence: 99%

Persistent entropy for separating topological features from noise in vietoris-rips complexes

2017

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Persistent homology studies the evolution of k-dimensional holes along a nested sequence of simplicial complexes (called a filtration). The set of bars (i.e. intervals) representing birth and death times of k-dimensional holes along such sequence is called the persistence barcode. k-Dimensional holes with short lifetimes are informally considered to be "topological noise", and those with long lifetimes are considered to be "topological features" associated to the filtration. Persistent entropy is defined as the Shannon entropy of the persistence barcode of a given filtration. In this paper we present new important properties of persistent entropy ofČech and Vietoris-Rips filtrations. Among the properties, we put a focus on the stability theorem that allows to use persistent entropy for comparing persistence barcodes. Later, we derive a simple method for separating topological noise from features in Vietoris-Rips filtrations.

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Section: Discussionmentioning

confidence: 99%

Persistent entropy for separating topological features from noise in vietoris-rips complexes

2017

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“…Algebraic topology provides a promising framework for extracting nonlinear features from finite metric spaces via the theory of persistent homology [17,26,28]. Persistent homology has solved a host of data-driven problems in disparate fields of science and engineering -examples include signal processing [30], proteomics [16], cosmology [32], sensor networks [13], molecular chemistry [34] and computer vision [23].…”

Section: Introductionmentioning

confidence: 99%

Persistence Paths and Signature Features in Topological Data Analysis

Chevyrev

Nanda

Oberhauser

2020

IEEE Trans. Pattern Anal. Mach. Intell.

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We introduce a new feature map for barcodes that arise in persistent homology computation. The main idea is to first realize each barcode as a path in a convenient vector space, and to then compute its path signature which takes values in the tensor algebra of that vector space. The composition of these two operations -barcode to path, path to tensor series -results in a feature map that has several desirable properties for statistical learning, such as universality and characteristicness, and achieves state-of-the-art results on common classification benchmarks.Persistence paths and signature features. Notwithstanding their usefulness for certain tasks, barcodes are notoriously unsuitable for standard statistical inference because 1 arXiv:1806.00381v2 [stat.ML]

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“…The closure of higher dimensional cliques has also been used for link prediction [22,23]. However, the clustering coefficient and its generalisations are insufficient to fully characterise the topology of a node neighbourhood, and therefore it is important to develop new Topological Data Analysis tools [24,25] that allow us to go beyond these simple metrics.…”

Section: Introductionmentioning

confidence: 99%

Beyond the clustering coefficient: A topological analysis of node neighbourhoods in complex networks

Kartun-Giles

Bianconi

2019

Chaos, Solitons & Fractals: X

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In Network Science node neighbourhoods, also called ego-centered networks have attracted large attention. In particular the clustering coefficient has been extensively used to measure their local cohesiveness. In this paper, we show how, given two nodes with the same clustering coefficient, the topology of their neighbourhoods can be significantly different, which demonstrates the need to go beyond this simple characterization. We perform a large scale statistical analysis of the topology of node neighbourhoods of real networks by first constructing their clique complexes, and then computing their Betti numbers. We are able to show significant differences between the topology of node neighbourhoods of real networks and the stochastic topology of null models of random simplicial complexes revealing local organisation principles of the node neighbourhoods.Moreover we observe that a large scale statistical analysis of the topological properties of node neighbourhoods is able to clearly discriminate between powerlaw networks, and planar road networks.

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Simplicial Models and Topological Inference in Biological Systems

Cited by 33 publications

References 28 publications

Persistent entropy for separating topological features from noise in vietoris-rips complexes

Persistent entropy for separating topological features from noise in vietoris-rips complexes

Persistence Paths and Signature Features in Topological Data Analysis

Beyond the clustering coefficient: A topological analysis of node neighbourhoods in complex networks

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