🍩 Database of Original & Non-Theoretical Uses of Topology

(found 10 matches in 0.00168s)

  3. Coverage Criterion in Sensor Networks Stable Under Perturbation (2014)

    Yasuaki Hiraoka, Genki Kusano
    Abstract To the coverage problem of sensor networks, V. de Silva and R. Ghrist (2007) developed several approaches based on (persistent) homology theory. Their criteria for the coverage are formulated on the Rips complexes constructed by the sensors, in which their locations are supposed to be fixed. However, the sensors are in general affected by perturbations (e.g., natural phenomena), and hence the stability of the coverage criteria should be also discussed. In this paper, we present a coverage theorem stable under perturbation. Furthermore, we also introduce a method of eliminating redundant cover after perturbation. The coverage theorem is derived by extending the Rips interleaving theorem studied by F. Chazal, V. de Silva, and S. Oudot (2013) into an appropriate relative version.

  4. Mapping Firms' Locations in Technological Space: A Topological Analysis of Patent Statistics (2020)

    Emerson G. Escolar, Yasuaki Hiraoka, Mitsuru Igami, Yasin Ozcan
    Abstract Where do firms innovate? Mapping their locations in technological space is difficult, because it is high dimensional and unstructured. We address this issue by using a method in computational topology called the Mapper algorithm, which combines local clustering with global reconstruction. We apply this method to a panel of 333 major firms’ patent portfolios in 1976–2005 across 430 technological areas. Results suggest the Mapper graph captures salient patterns in firms’ patenting histories, and our measures of their uniqueness (the length of “flares”) are correlated with firms’ financial performances in a statistically and economically significant manner. We then compare this approach with a widely used clustering method by Jaffe (1989) to highlight additional findings.

  5. Protein-Folding Analysis Using Features Obtained by Persistent Homology (2020)

    Takashi Ichinomiya, Ippei Obayashi, Yasuaki Hiraoka
    Abstract Understanding the protein-folding process is an outstanding issue in biophysics; recent developments in molecular dynamics simulation have provided insights into this phenomenon. However, the large freedom of atomic motion hinders the understanding of this process. In this study, we applied persistent homology, an emerging method to analyze topological features in a data set, to reveal protein-folding dynamics. We developed a new, to our knowledge, method to characterize the protein structure based on persistent homology and applied this method to molecular dynamics simulations of chignolin. Using principle component analysis or nonnegative matrix factorization, our analysis method revealed two stable states and one saddle state, corresponding to the native, misfolded, and transition states, respectively. We also identified an unfolded state with slow dynamics in the reduced space. Our method serves as a promising tool to understand the protein-folding process.

  6. Persistent Homology and Many-Body Atomic Structure for Medium-Range Order in the Glass (2015)

    Takenobu Nakamura, Yasuaki Hiraoka, Akihiko Hirata, Emerson G. Escolar, Yasumasa Nishiura
    Abstract The characterization of the medium-range (MRO) order in amorphous materials and its relation to the short-range order is discussed. A new topological approach to extract a hierarchical structure of amorphous materials is presented, which is robust against small perturbations and allows us to distinguish it from periodic or random configurations. This method is called the persistence diagram (PD) and introduces scales to many-body atomic structures to facilitate size and shape characterization. We first illustrate the representation of perfect crystalline and random structures in PDs. Then, the MRO in amorphous silica is characterized using the appropriate PD. The PD approach compresses the size of the data set significantly, to much smaller geometrical summaries, and has considerable potential for application to a wide range of materials, including complex molecular liquids, granular materials, and metallic glasses.

  7. A Topological Measurement of Protein Compressibility (2015)

    Marcio Gameiro, Yasuaki Hiraoka, Shunsuke Izumi, Miroslav Kramar, Konstantin Mischaikow, Vidit Nanda
    Abstract In this paper we partially clarify the relation between the compressibility of a protein and its molecular geometric structure. To identify and understand the relevant topological features within a given protein, we model its molecule as an alpha filtration and hence obtain multi-scale insight into the structure of its tunnels and cavities. The persistence diagrams of this alpha filtration capture the sizes and robustness of such tunnels and cavities in a compact and meaningful manner. From these persistence diagrams, we extract a measure of compressibility derived from those topological features whose relevance is suggested by physical and chemical properties. Due to recent advances in combinatorial topology, this measure is efficiently and directly computable from information found in the Protein Data Bank (PDB). Our main result establishes a clear linear correlation between the topological measure and the experimentally-determined compressibility of most proteins for which both PDB information and experimental compressibility data are available. Finally, we establish that both the topological measurement and the linear correlation are stable with respect to small perturbations in the input data, such as those arising from experimental errors in compressibility and X-ray crystallography experiments.

  8. Hepatic Tumor Classification Using Texture and Topology Analysis of Non-Contrast-Enhanced Three-Dimensional T1-Weighted MR Images With a Radiomics Approach (2019)

    Asuka Oyama, Yasuaki Hiraoka, Ippei Obayashi, Yusuke Saikawa, Shigeru Furui, Kenshiro Shiraishi, Shinobu Kumagai, Tatsuya Hayashi, Jun’ichi Kotoku
    Abstract The purpose of this study is to evaluate the accuracy for classification of hepatic tumors by characterization of T1-weighted magnetic resonance (MR) images using two radiomics approaches with machine learning models: texture analysis and topological data analysis using persistent homology. This study assessed non-contrast-enhanced fat-suppressed three-dimensional (3D) T1-weighted images of 150 hepatic tumors. The lesions included 50 hepatocellular carcinomas (HCCs), 50 metastatic tumors (MTs), and 50 hepatic hemangiomas (HHs) found respectively in 37, 23, and 33 patients. For classification, texture features were calculated, and also persistence images of three types (degree 0, degree 1 and degree 2) were obtained for each lesion from the 3D MR imaging data. We used three classification models. In the classification of HCC and MT (resp. HCC and HH, HH and MT), we obtained accuracy of 92% (resp. 90%, 73%) by texture analysis, and the highest accuracy of 85% (resp. 84%, 74%) when degree 1 (resp. degree 1, degree 2) persistence images were used. Our methods using texture analysis or topological data analysis allow for classification of the three hepatic tumors with considerable accuracy, and thus might be useful when applied for computer-aided diagnosis with MR images.

  9. Hierarchical Structures of Amorphous Solids Characterized by Persistent Homology (2016)

    Yasuaki Hiraoka, Takenobu Nakamura, Akihiko Hirata, Emerson G. Escolar, Kaname Matsue, Yasumasa Nishiura
    Abstract This article proposes a topological method that extracts hierarchical structures of various amorphous solids. The method is based on the persistence diagram (PD), a mathematical tool for capturing shapes of multiscale data. The input to the PDs is given by an atomic configuration and the output is expressed as 2D histograms. Then, specific distributions such as curves and islands in the PDs identify meaningful shape characteristics of the atomic configuration. Although the method can be applied to a wide variety of disordered systems, it is applied here to silica glass, the Lennard-Jones system, and Cu-Zr metallic glass as standard examples of continuous random network and random packing structures. In silica glass, the method classified the atomic rings as short-range and medium-range orders and unveiled hierarchical ring structures among them. These detailed geometric characterizations clarified a real space origin of the first sharp diffraction peak and also indicated that PDs contain information on elastic response. Even in the Lennard-Jones system and Cu-Zr metallic glass, the hierarchical structures in the atomic configurations were derived in a similar way using PDs, although the glass structures and properties substantially differ from silica glass. These results suggest that the PDs provide a unified method that extracts greater depth of geometric information in amorphous solids than conventional methods.

  10. Understanding Diffraction Patterns of Glassy, Liquid and Amorphous Materials via Persistent Homology Analyses (2019)

    Yohei Onodera, Shinji Kohara, Shuta Tahara, Atsunobu Masuno, Hiroyuki Inoue, Motoki Shiga, Akihiko Hirata, Koichi Tsuchiya, Yasuaki Hiraoka, Ippei Obayashi, Koji Ohara, Akitoshi Mizuno, Osami Sakata
    Abstract The structure of glassy, liquid, and amorphous materials is still not well understood, due to the insufficient structural information from diffraction data. In this article, attempts are made to understand the origin of diffraction peaks, particularly of the first sharp diffraction peak (FSDP, Q1), the principal peak (PP, Q2), and the third peak (Q3), observed in the measured diffraction patterns of disordered materials whose structure contains tetrahedral motifs. It is confirmed that the FSDP (Q1) is not a signature of the formation of a network, because an FSDP is observed in tetrahedral molecular liquids. It is found that the PP (Q2) reflects orientational correlations of tetrahedra. Q3, that can be observed in all disordered materials, even in common liquid metals, stems from simple pair correlations. Moreover, information on the topology of disordered materials was revealed by utilizing persistent homology analyses. The persistence diagram of silica (SiO2) glass suggests that the shape of rings in the glass is similar not only to those in the crystalline phase with comparable density (α-cristobalite), but also to rings present in crystalline phases with higher density (α-quartz and coesite); this is thought to be the signature of disorder. Furthermore, we have succeeded in revealing the differences, in terms of persistent homology, between tetrahedral networks and tetrahedral molecular liquids, and the difference/similarity between liquid and amorphous (glassy) states. Our series of analyses demonstrated that a combination of diffraction data and persistent homology analyses is a useful tool for allowing us to uncover structural features hidden in halo pattern of disordered materials.