🍩 Database of Original & NonTheoretical Uses of Topology
(found 10 matches in 0.001928s)


Persistent Homology in Cosmic Shear: Constraining Parameters With Topological Data Analysis (2021)
Sven Heydenreich, Benjamin Brück, Joachim HarnoisDérapsAbstract
In recent years, cosmic shear has emerged as a powerful tool for studying the statistical distribution of matter in our Universe. Apart from the standard twopoint correlation functions, several alternative methods such as peak count statistics offer competitive results. Here we show that persistent homology, a tool from topological data analysis, can extract more cosmological information than previous methods from the same data set. For this, we use persistent Betti numbers to efficiently summarise the full topological structure of weak lensing aperture mass maps. This method can be seen as an extension of the peak count statistics, in which we additionally capture information about the environment surrounding the maxima. We first demonstrate the performance in a mock analysis of the KiDS+VIKING450 data: We extract the Betti functions from a suite of \textlessi\textgreaterN\textlessi/\textgreaterbody simulations and use these to train a Gaussian process emulator that provides rapid model predictions; we next run a Markov chain Monte Carlo analysis on independent mock data to infer the cosmological parameters and their uncertainties. When comparing our results, we recover the input cosmology and achieve a constraining power on that is 3% tighter than that on peak count statistics. Performing the same analysis on 100 deg\textlesssup\textgreater2\textlesssup/\textgreater of \textlessi\textgreaterEuclid\textlessi/\textgreaterlike simulations, we are able to improve the constraints on \textlessi\textgreaterS\textlessi/\textgreater\textlesssub\textgreater8\textlesssub/\textgreater and Ω\textlesssub\textgreaterm\textlesssub/\textgreater by 19% and 12%, respectively, while breaking some of the degeneracy between \textlessi\textgreaterS\textlessi/\textgreater\textlesssub\textgreater8\textlesssub/\textgreater and the dark energy equation of state. To our knowledge, the methods presented here are the most powerful topological tools for constraining cosmological parameters with lensing data. 
Persistent Homology of Geospatial Data: A Case Study With Voting (2021)
Michelle Feng, Mason A. PorterAbstract
A crucial step in the analysis of persistent homology is the transformation of data into an appropriate topological object (which, in our case, is a simplicial complex). Software packages for computing persistent homology typically construct VietorisRips or other distancebased simplicial complexes on point clouds because they are relatively easy to compute. We investigate alternative methods of constructing simplicial complexes and the effects of making associated choices during simplicialcomplex construction on the output of persistenthomology algorithms. We present two new methods for constructing simplicial complexes from twodimensional geospatial data (such as maps). We apply these methods to a California precinctlevel voting data set, and we thereby demonstrate that our new constructions can capture geometric characteristics that are missed by distancebased constructions. Our new constructions can thus yield more interpretable persistence modules and barcodes for geospatial data. In particular, they are able to distinguish shortpersistence features that occur only for a narrow range of distance scales (e.g., voting patterns in densely populated cities) from shortpersistence noise by incorporating information about other spatial relationships between regions. 
The (Homological) Persistence of Gerrymandering (2021)
Moon Duchin, Tom Needham, Thomas WeighillAbstract
\textlessp style='textindent:20px;'\textgreaterWe apply persistent homology, the dominant tool from the field of topological data analysis, to study electoral redistricting. We begin by combining geographic and electoral data from a districting plan to produce a persistence diagram. Then, to see beyond a particular plan and understand the possibilities afforded by the choices made in redistricting, we build methods to visualize and analyze large ensembles of alternative plans. Our detailed case studies use zerodimensional homology (persistent components) of filtered graphs constructed from voting data to analyze redistricting in Pennsylvania and North Carolina. We find that, across large ensembles of partitions, the features cluster in the persistence diagrams in a way that corresponds strongly to geographic location, so that we can construct an average diagram for an ensemble, with each point identified with a geographical region. Using this localization lets us produce zonings of each state at Congressional, state Senate, and state House scales, show the regional nonuniformity of election shifts, and identify attributes of partitions that tend to correspond to partisan advantage.\textless/p\textgreater\textlessp style='textindent:20px;'\textgreaterThe methods here are set up to be broadly applicable to the use of TDA on large ensembles of data. Many studies will benefit from interpretable summaries of large sets of samples or simulations, and the work here on localization and zoning will readily generalize to other partition problems, which are abundant in scientific applications. For the mathematically and politically rich problem of redistricting in particular, TDA provides a powerful and elegant summarization tool whose findings will be useful for practitioners.\textless/p\textgreater 
Loops Abound in the Cosmic Microwave Background: A \$4\sigma\$ Anomaly on SuperHorizon Scales (2021)
Pratyush PranavAbstract
We present a topological analysis of the temperature fluctuation maps from the \emph\Planck 2020\ Data release 4 (DR4) based on the \texttt\NPIPE\ data processing pipeline. For comparison, we also present the topological characteristics of the maps from \emph\Planck 2018\ Data release 3 (DR3). We perform our analysis in terms of the homology characteristics of the maps, invoking relative homology to account for analysis in the presence of masks. We perform our analysis for a range of smoothing scales spanning sub and superhorizon scales corresponding to \$FWHM = 5', 10', 20', 40', 80', 160', 320', 640'\$. Our main result indicates a significantly anomalous behavior of the loops in the observed maps compared to simulations that are modeled as isotopic and homogeneous Gaussian random fields. Specifically, we observe a \$4\sigma\$ deviation between the observation and simulations in the number of loops at \$FWHM = 320'\$ and \$FWHM = 640'\$, corresponding to superhorizon scales of \$5\$ degrees and larger. In addition, we also notice a mildly significant deviation at \$2\sigma\$ for all the topological descriptors for almost all the scales analyzed. Our results show a consistency across different data releases, and therefore, the anomalous behavior deserves a careful consideration regarding its origin and ramifications. Disregarding the unlikely source of the anomaly being instrumental systematics, the origin of the anomaly may be genuinely astrophysical  perhaps due to a yet unresolved foreground, or truly primordial in nature. Given the nature of the topological descriptors, that potentially encodes information of all orders, nonGaussianities, of either primordial or latetype nature, may be potential candidates. Alternate possibilities include the Universe admitting a nontrivial global topology, including effects induced by largescale topological defects. 
Topological Data Analysis of SingleCell HiC Contact Maps (2020)
Mathieu Carrière, Raúl RabadánAbstract
Due to recent breakthroughs in highthroughput sequencing, it is now possible to use chromosome conformation capture (CCC) to understand the three dimensional conformation of DNA at the whole genome level, and to characterize it with the socalled contact maps. This is very useful since many biological processes are correlated with DNA folding, such as DNA transcription. However, the methods for the analysis of such conformations are still lacking mathematical guarantees and statistical power. To handle this issue, we propose to use the Mapper, which is a standard tool of Topological Data Analysis (TDA) that allows one to efficiently encode the inherent continuity and topology of underlying biological processes in data, in the form of a graph with various features such as branches and loops. In this article, we show how recent statistical techniques developed in TDA for the Mapper algorithm can be extended and leveraged to formally define and statistically quantify the presence of topological structures coming from biological phenomena, such as the cell cyle, in datasets of CCC contact maps. 
Identification of Topological Network Modules in Perturbed Protein Interaction Networks (2017)
Mihaela E. Sardiu, Joshua M. Gilmore, Brad Groppe, Laurence Florens, Michael P. WashburnAbstract
Biological networks consist of functional modules, however detecting and characterizing such modules in networks remains challenging. Perturbing networks is one strategy for identifying modules. Here we used an advanced mathematical approach named topological data analysis (TDA) to interrogate two perturbed networks. In one, we disrupted the S. cerevisiae INO80 protein interaction network by isolating complexes after protein complex components were deleted from the genome. In the second, we reanalyzed previously published data demonstrating the disruption of the human Sin3 network with a histone deacetylase inhibitor. Here we show that disrupted networks contained topological network modules (TNMs) with shared properties that mapped onto distinct locations in networks. We define TMNs as proteins that occupy close network positions depending on their coordinates in a topological space. TNMs provide new insight into networks by capturing proteins from different categories including proteins within a complex, proteins with shared biological functions, and proteins disrupted across networks. 
Topological Data Analysis of Contagion Maps for Examining Spreading Processes on Networks (2015)
Dane Taylor, Florian Klimm, Heather A. Harrington, Miroslav Kramár, Konstantin Mischaikow, Mason A. Porter, Peter J. MuchaAbstract
Social and biological contagions are influenced by the spatial embeddedness of networks. Historically, many epidemics spread as a wave across part of the Earth’s surface; however, in modern contagions longrange edges—for example, due to airline transportation or communication media—allow clusters of a contagion to appear in distant locations. Here we study the spread of contagions on networks through a methodology grounded in topological data analysis and nonlinear dimension reduction. We construct ‘contagion maps’ that use multiple contagions on a network to map the nodes as a point cloud. By analysing the topology, geometry and dimensionality of manifold structure in such point clouds, we reveal insights to aid in the modelling, forecast and control of spreading processes. Our approach highlights contagion maps also as a viable tool for inferring lowdimensional structure in networks. 
Persistent Homology for Path Planning in Uncertain Environments (2015)
S. Bhattacharya, R. Ghrist, V. KumarAbstract
We address the fundamental problem of goaldirected path planning in an uncertain environment represented as a probability (of occupancy) map. Most methods generally use a threshold to reduce the grayscale map to a binary map before applying offtheshelf techniques to find the best path. This raises the somewhat illposed question, what is the right (optimal) value to threshold the map? We instead suggest a persistent homology approach to the problema topological approach in which we seek the homology class of trajectories that is most persistent for the given probability map. In other words, we want the class of trajectories that is free of obstacles over the largest range of threshold values. In order to make this problem tractable, we use homology in ℤ2 coefficients (instead of the standard ℤ coefficients), and describe how graph searchbased algorithms can be used to find trajectories in different homology classes. Our simulation results demonstrate the efficiency and practical applicability of the algorithm proposed in this paper.paper. 
Reconceiving the Hippocampal Map as a Topological Template (2014)
Yuri Dabaghian, Vicky L. Brandt, Loren M. FrankAbstract
The role of the hippocampus in spatial cognition is incontrovertible yet controversial. Place cells, initially thought to be locationspecifiers, turn out to respond promiscuously to a wide range of stimuli. Here we test the idea, which we have recently demonstrated in a computational model, that the hippocampal place cells may ultimately be interested in a space's topological qualities (its connectivity) more than its geometry (distances and angles); such higherorder functioning would be more consistent with other known hippocampal functions. We recorded place cell activity in rats exploring morphing linear tracks that allowed us to dissociate the geometry of the track from its topology. The resulting place fields preserved the relative sequence of places visited along the track but did not vary with the metrical features of the track or the direction of the rat's movement. These results suggest a reinterpretation of previous studies and new directions for future experiments.