Random graph and persistent homology

• Project/Area Number: 26610025
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Kyushu University
• Principal Investigator: Shirai Tomoyuki 九州大学, マス・フォア・インダストリ研究所, 教授 (70302932)
• Co-Investigator(Kenkyū-buntansha): 平岡 裕章 東北大学, 原子分子材料科学高等研究機構, 教授 (10432709) ; 樋口 雄介 昭和大学, 教養部, 講師 (20286842)
• Project Period (FY): 2014-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2014: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: パーシステントホモロジー / ランダムグラフ / ランダム複体 / エルデシュ・レンニグラフ / リニアル・メシュラム複体 / ランダムセル複体 / エルデシュ・レンニランダムグラフ / クリーク複体

Outline of Final Research Achievements

Persistent homology is known as a tool of topological data analysis and we apply it to the theory of random graphs. We emphasize the viewpoint of persistent homology to apply it to random complexes. This study opens a way to analyze a family of random complexes as well as random complexes themselves, and also this enables us to develop the theory of random complexes and provides new viewpoints on this area.

Research Products

• [Journal Article] Tutte polynomials and random-cluster models in Bernoulli cell complexes (2016)
  • Author(s): Yasuaki Hiraoka and Tomoyuki Shirai
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: B59 Pages: 289-304
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] ランダム複体とパーシステントホモロジー (2015)
  • Author(s): 白井朋之
  • Journal Title: 京都大学数理解析研講究録 Volume: 1986
  • Open Access / Acknowledgement Compliant
• [Journal Article] ランダム複体とパーシステントホモロジー (2015)
  • Author(s): 白井朋之・平岡裕章
  • Journal Title: 京都大学数理解析研究所講究録 Volume: 未定
  • Open Access
• [Presentation] Persistent homology of random complexes built over point processes (2016)
  • Author(s): Tomoyuki Shirai
  • Organizer: Workshop on "Geometry and Probability"
  • Place of Presentation: RIMS, Kyoto University
  • Year and Date: 2016-10-24
  • Int'l Joint Research / Invited
• [Presentation] Persistent homology and minimum spanning acycle for certain random complexes (2016)
  • Author(s): Tomoyuki Shirai
  • Organizer: Workshop on "High-Dimensional Expanders 2016"
  • Place of Presentation: Les Diablerets, Switzerland
  • Year and Date: 2016-06-19
  • Int'l Joint Research / Invited
• [Presentation] ランダム複体とパーシステントホモロジー (2015)
  • Author(s): 白井朋之
  • Organizer: 平成27年度RIMS共同研究「デザイン，符号，グラフおよびその周辺」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2015-07-08
  • Invited
• [Presentation] Persistent homology and minimum spanning acycle for random simplicial complexes (2015)
  • Author(s): Tomoyuki Shirai
  • Organizer: Group Representations in Dynamical Systems and Geometry
  • Place of Presentation: CIRM, Marseille, France
  • Year and Date: 2015-07-03
  • Int'l Joint Research / Invited
• [Presentation] Lifetime Sum of Persistent Homology and Minimum Spanning Acycles in Random Simplicial Complexes (2015)
  • Author(s): Tomoyuki Shirai
  • Organizer: Topological Data Analysis on Materials Science
  • Place of Presentation: 東北大学AIMR
  • Year and Date: 2015-02-20
  • Invited
• [Remarks] Tomoyuki Shirai
  • URL: http://imi.kyushu-u.ac.jp/~shirai/