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A Differential Geometric Research on the Construction of Highly Connected Graphs Applicable to Big Data Analysis

Research Project

Project/Area Number 19K23411
Research Category

Grant-in-Aid for Research Activity Start-up

Allocation TypeMulti-year Fund
Review Section 0201:Algebra, geometry, analysis, applied mathematics,and related fields
Research InstitutionShimane University (2020-2021)
Research Institute for Humanity and Nature (2019)

Principal Investigator

Yamada Taiki  島根大学, 学術研究院理工学系, 助教 (00847270)

Project Period (FY) 2019-08-30 – 2022-03-31
Project Status Completed (Fiscal Year 2021)
Budget Amount *help
¥2,470,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥570,000)
Fiscal Year 2020: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2019: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywordsグラフ理論 / リッチ曲率 / 連結度 / ラプラシアン / 単体的複体 / 幾何解析 / グラフラプラシアン / 微分幾何学
Outline of Research at the Start

近年,ビッグデータの整備が進み,より複雑なグラフを解析する必要がある.こうした複雑なグラフに対して,より効率的に解析するため,局所的な計算で大域的性質を得ることができる微分幾何学的手法を構成することが本研究の目的である.その際,必要な概念はリッチ曲率と呼ばれるもので,これはリーマン幾何学において,多様体の構造を理解する上で重要な役割を果たすものである.このリッチ曲率をグラフ上に拡張したものと,ビッグデータの解析にも用いられているグラフの結びつきの強さを表す連結度との関係を明らかにすることで,ビッグデータ解析にも応用可能な高連結度グラフを構成方法を確立する.

Outline of Final Research Achievements

The purpose of this research is to establish a differential geometrical method that can construct graphs that strengthen graph connectivity, an invariant that indicates the strength of graph ties, and we challenge the pioneering theory of constructing new algorithms.
The connectivity of graphs has been widely applied to real-world problems such as fault-tolerant network design problems, but recently, with the development of big data, it has become necessary to deal with larger graphs. In Riemannian geometry, we focused on Ricci curvature, which can obtain the structure of manifolds from local calculations.
The results of this research have been summarized in three papers, all of which have been published in international journals.

Academic Significance and Societal Importance of the Research Achievements

本研究は,申請者の構築した有向グラフ上の理論やコホモロジー論を用いることで,微分幾何学の概念であるリッチ曲率と組合せ論の概念である連結度を結びつける接合的研究に位置付けられるため,双方の学問分野において波及効果を及ぼす.また,ビッグデータの整備が進んでいる昨今において,効率的な解析手法を提案する本研究は学術領域だけでなく,実社会にも大きなインパクトを与える.

Report

(4 results)
  • 2021 Annual Research Report   Final Research Report ( PDF )
  • 2020 Research-status Report
  • 2019 Research-status Report
  • Research Products

    (8 results)

All 2021 2020 2019

All Journal Article (5 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 5 results,  Open Access: 1 results) Presentation (3 results) (of which Invited: 1 results)

  • [Journal Article] Maximal diameter theorem for directed graphs of positive Ricci curvature2021

    • Author(s)
      Ryunosuke Ozawa, Yohei Sakurai, Taiki Yamada
    • Journal Title

      Communications in Analysis and Geometry

      Volume: -

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Geometric and spectral properties of directed graphs under a lower Ricci curvature bound2020

    • Author(s)
      Ozawa Ryunosuke, Sakurai Yohei, Yamada Taiki
    • Journal Title

      Calculus of Variations and Partial Differential Equations

      Volume: 59 Issue: 4 Pages: 142-177

    • DOI

      10.1007/s00526-020-01809-2

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] AN ESTIMATE OF THE FIRST NON-ZERO EIGENVALUE OF THE LAPLACIAN BY THE RICCI CURVATURE ON EDGES OF GRAPHS2020

    • Author(s)
      Taiki Yamada
    • Journal Title

      Osaka Journal of Mathematics

      Volume: 57 Issue: 1 Pages: 151-163

    • DOI

      10.18910/73742

    • NAID

      120006786537

    • ISSN
      00306126
    • URL

      https://ocu-omu.repo.nii.ac.jp/records/2010767

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] A Construction of Graphs with Positive Ricci Curvature2020

    • Author(s)
      Taiki Yamada
    • Journal Title

      Kyushu Journal of Mathematics

      Volume: in printing

    • NAID

      130007956239

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Relation between Combinatorial Ricci Curvature and Lin-Lu-Yau's Ricci Curvature on Cell Complexes2019

    • Author(s)
      Kazuyoshi Watanabe, Taiki Yamada
    • Journal Title

      Tokyo Journal of Mathematics

      Volume: in printing

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Presentation] The lower bound of the eigenvalue of the Laplacian on simplicial complex by the Ricci curvature2020

    • Author(s)
      山田大貴
    • Organizer
      日本数学会秋季総合分科会
    • Related Report
      2020 Research-status Report
  • [Presentation] Construction of the stable transaction network by using graphs with positive Ricci curvature2019

    • Author(s)
      山田大貴
    • Organizer
      地球研セミナー
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] Laplacian Comparison by the Ricci Curvature on Directed Graphs2019

    • Author(s)
      山田大貴
    • Organizer
      数理計画問題に対する理論とアルゴリズムの研究
    • Related Report
      2019 Research-status Report

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Published: 2019-09-03   Modified: 2023-01-30  

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