Persistent homology for random fields and complexes

Adler, Robert J.; Bobrowski, Omer; Borman, Matthew Strom; Weinberger, Shmuel

doi:10.1214/10-imscoll609

Cited by 83 publications

(110 citation statements)

References 34 publications

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“…Statistical properties of the fluctuations in the gas properties are strongly non-Gaussian because of widespread filamentary and small-scale planar structures. Such features cannot be captured by second-order correlation functions (or their equivalent, power spectra) and require other tools sensitive to all statistical moments of the random field, such as Minkowski functionals (e.g., Wilkin et al 2007;Makarenko et al 2015, and references therein) and topological data analysis (Adler et al 2010;Edelsbrunner 2014). However, careful correlation analysis remains a necessary first step in the exploration of statistical properties of random fields.…”

Section: Discussionmentioning

confidence: 99%

Supernova-regulated ISM. V. Space and Time Correlations

Hollins

Sarson

Shukurov

et al. 2017

ApJ

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We apply correlation analysis to random fields in numerical simulations of the supernova-driven interstellar medium (ISM) with the magnetic field produced by dynamo action. We solve the magnetohydrodynamic (MHD) equations in a shearing Cartesian box representing a local region of the ISM, subject to thermal and kinetic energy injection by supernova explosions, and parameterized, optically thin radiative cooling. We consider the cold, warm, and hot phases of the ISM separately; the analysis mostly considers the warm gas, which occupies the bulk of the domain. Various physical variables have different correlation lengths in the warm phase: 40, 50, and 60 pc for the random magnetic field, density, and velocity, respectively, in the midplane. The correlation time of the random velocity is comparable to the eddy turnover time, about 10 year 7 , although it may be shorter in regions with a higher star formation rate. The random magnetic field is anisotropic, with the standard deviations of its components b b b x y z having approximate ratios 0.5 0.6 0.6 in the midplane. The anisotropy is attributed to the global velocity shear from galactic differential rotation and locally inhomogeneous outflow to the galactic halo. The correlation length of Faraday depth along the z axis, 120 pc, is greater than for electron density, 60 90 pc -, and the vertical magnetic field, 60 pc. Such comparisons may be sensitive to the orientation of the line of sight. Uncertainties of the structure functions of synchrotron intensity rapidly increase with the scale. This feature is hidden in a power spectrum analysis, which can undermine the usefulness of power spectra for detailed studies of interstellar turbulence.

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Section: Discussionmentioning

confidence: 99%

Supernova-regulated ISM. V. Space and Time Correlations

Hollins

Sarson

Shukurov

et al. 2017

ApJ

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“…Its aim, achieved through topological filtration, is to isolate significant properties of a random field that can be used to simplify it and thus to make it amenable to analysis, comparison and statistical inference. Rigorous definitions of the Betti numbers and related concepts can be found in Adler et al (2010); Edelsbrunner & Harer (2010); Adler & Taylor (2011) and Edelsbrunner (2014) while Park et al (2013) and Pranav et al (2017) provide useful and less formal expositions. Here we briefly present the basics at an intuitive level.…”

Section: Topological Data Analysismentioning

confidence: 99%

“…However, the morphology of an intermittent random field is only one of its aspects. More subtle but no less essential features are revealed by topological filtration, which characterises statistical properties of the extrema of the random field and connectivity of its isosurfaces (Carlsson 2009;Adler et al 2010;Adler & Taylor 2011;Edelsbrunner 2014). These features are described in terms of the Betti numbers, β 0 , β 1 and β 2 ; in a space of a dimension d, there are d Betti numbers.…”

Section: Introductionmentioning

confidence: 99%

Topological signatures of interstellar magnetic fields – I. Betti numbers and persistence diagrams

Makarenko

Shukurov

Henderson

et al. 2018

Monthly Notices of the Royal Astronomical Society

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The interstellar medium (ISM) is a magnetised system in which transonic or supersonic turbulence is driven by supernova explosions. This leads to the production of intermittent, filamentary structures in the ISM gas density, whilst the associated dynamo action also produces intermittent magnetic fields. The traditional theory of random functions, restricted to second-order statistical moments (or power spectra), does not adequately describe such systems. We apply topological data analysis (TDA), sensitive to all statistical moments and independent of the assumption of Gaussian statistics, to the gas density fluctuations in a magnetohydrodynamic (MHD) simulation of the multi-phase ISM. This simulation admits dynamo action, so produces physically realistic magnetic fields. The topology of the gas distribution, with and without magnetic fields, is quantified in terms of Betti numbers and persistence diagrams. Like the more standard correlation analysis, TDA shows that the ISM gas density is sensitive to the presence of magnetic fields. However, TDA gives us important additional information that cannot be obtained from correlation functions. In particular, the Betti numbers per correlation cell are shown to be physically informative. Magnetic fields make the ISM more homogeneous, reducing the abundance of both isolated gas clouds and cavities, with a stronger effect on the cavities. Remarkably, the modification of the gas distribution by magnetic fields is captured by the Betti numbers even in regions more than 300 pc from the midplane, where the magnetic field is weaker and correlation analysis fails to detect any signatures of magnetic effects.

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“…However, it is unclear which measure is appropriate for network comparison. Instead of trying to find one particular characteristic of network at a given scale, one can also look at the overall change of topological features through persistent homology [5,6,7]. In the persistent homology, the topological features such as the connected components and circles of the network are tabulated in terms of the algebraic form known as Betti numbers.…”

Section: Introductionmentioning

confidence: 99%

Computing the Shape of Brain Networks Using Graph Filtration and Gromov-Hausdorff Metric

Lee

Chung

Kang

et al. 2011

Lecture Notes in Computer Science

115

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Abstract.The difference between networks has been often assessed by the difference of global topological measures such as the clustering coefficient, degree distribution and modularity. In this paper, we introduce a new framework for measuring the network difference using the GromovHausdorff (GH) distance, which is often used in shape analysis. In order to apply the GH distance, we define the shape of the brain network by piecing together the patches of locally connected nearest neighbors using the graph filtration. The shape of the network is then transformed to an algebraic form called the single linkage matrix. The single linkage matrix is subsequently used in measuring network differences using the GH distance. As an illustration, we apply the proposed framework to compare the FDG-PET based functional brain networks out of 24 attention deficit hyperactivity disorder (ADHD) children, 26 autism spectrum disorder (ASD) children and 11 pediatric control subjects.

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Persistent homology for random fields and complexes

Cited by 83 publications

References 34 publications

Supernova-regulated ISM. V. Space and Time Correlations

Supernova-regulated ISM. V. Space and Time Correlations

Topological signatures of interstellar magnetic fields – I. Betti numbers and persistence diagrams

Computing the Shape of Brain Networks Using Graph Filtration and Gromov-Hausdorff Metric

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