Spectral analysis of complex networks and modeling methods by statistics

• Project/Area Number: 23654046
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Tohoku University
• Principal Investigator: OBATA Nobuaki 東北大学, 情報科学研究科, 教授 (10169360)
• Co-Investigator(Renkei-kenkyūsha): HORA Akihito 北海道大学, 大学院・理学研究院, 教授 (10212200) ; KONNO Norio 横浜国立大学, 大学院・工学研究院, 教授 (80205575)
• Project Period (FY): 2011 – 2013
• Project Status: Completed (Fiscal Year 2013)
• Budget Amount: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2013: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2012: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2011: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: ネットワーク / 量子確率 / 量子ウォーク / ランダムグラフ / スペクトル / 局在化 / 有向グラフ / 固有値分布

Research Abstract

(1) Spectral analysis of networks: The asymptotic spectral distribution of the distance-k graph of the cartesian product graph is derived as a combinatorial central limit theorem. The asymptotic spectral distribution of weighted hypercubes are described in terms of q-deformed Hermite polynomials. The spectral distribution is completely determined for the Manhattan product of directed paths of small orders, and thier asymptotic behaviors are obtained in the infinite length.
(2) Statistical and asymptotic behaviors of dynamics for the modeling methods: A quantum probabilistic method is applied to classical random walks for reconstructing transition probability and recurrence probability. A necessary and sufficient condition is obtained for a quantum walk (Grover walk) on a spidernet in order to exhibit the localization.

Research Products

• [Journal Article] The Karlin-McGregor formula for paths connected with a clique (2013)
  • Author(s): Nobuaki Obata
  • Journal Title: Prob. Math. Stat Volume: 33(2) Pages: 451-466
  • Peer Reviewed
• [Journal Article] Asymptotic spectral distributions of distance-k graphs of Cartesian product graphs (2013)
  • Author(s): Yuji Hibino, Hun Hee Lee, and Nobuaki Obata
  • Journal Title: Colloq. Math. Volume: 132 Issue: 1 Pages: 35-51
  • DOI: 10.4064/cm132-1-4
  • Peer Reviewed
• [Journal Article] Distance-k graphs of hypercube and q-Hermite polynomials (2013)
  • Author(s): Hun Hee Lee and Nobuaki Obata
  • Journal Title: Infin. Dimens. Anal. Quantum Probab. Relat. Top. Volume: 16 Issue: 02 Pages: 1350011-1350011
  • DOI: 10.1142/s0219025713500112
  • Peer Reviewed
• [Journal Article] Localization of the Grover walks on spidernets and free Meixner laws (2013)
  • Author(s): Norio Konno, Nobuaki Obata, and Etsuo Segawa
  • Journal Title: Comm. Math. Phys. Volume: 322 Issue: 3 Pages: 667-695
  • DOI: 10.1007/s00220-013-1742-x
  • Peer Reviewed
• [Journal Article] Spectra of Manhattan Products of Directed Paths <i>P</i>_<i>n</i>#<i>P</i>₂ (2012)
  • Author(s): Nobuaki Obata
  • Journal Title: Interdisciplinary Information Sciences Volume: 18 Issue: 1 Pages: 43-54
  • DOI: 10.4036/iis.2012.43
  • NAID: 130001916189
  • ISSN: 1340-9050, 1347-6157
  • Peer Reviewed
• [Journal Article] One-mode interacting Fock spaces and random walks on graphs (2012)
  • Author(s): Nobuaki Obata
  • Journal Title: Stochastics Volume: 84 Issue: 2-3 Pages: 383-392
  • DOI: 10.1080/17442508.2010.550919
  • Peer Reviewed
• [Journal Article] Manhattan products of digraphs : characteristic polynomials and examples (2012)
  • Author(s): Nobuaki Obata
  • Journal Title: Ann. Funct. Anal Volume: 3(2) Pages: 135-143
  • Peer Reviewed
• [Journal Article] Asymptotic spectral distributions of distance k-graphs of large-dimensional hypercubes (2012)
  • Author(s): Nobuaki Obata
  • Journal Title: Banach Center Publications Volume: 96 Pages: 287-292
  • DOI: 10.4064/bc96-0-20
  • Peer Reviewed
• [Journal Article] Manhattan Products of Digraphs: Characteristic polynomials and examples (2012)
  • Author(s): Nobuaki Obata
  • Journal Title: Ann. Funct. Anal. Volume: 3 Pages: 135-143
  • Peer Reviewed
• [Journal Article] Asymptotic spectral distributions of distance ... (2012)
  • Author(s): Nobuaki Obata
  • Journal Title: to appear in Banach Center Publications Volume: 未定
  • Peer Reviewed
• [Journal Article] Manhattan product of digraphs (2012)
  • Author(s): Nobuaki Obata
  • Journal Title: to appear in RIMS Kokyuroku Volume: 未定
• [Journal Article] Markov product of positive definite kernels and applications to Q-matrices of graph products (2011)
  • Author(s): Nobuaki Obata
  • Journal Title: Colloq. Math Volume: 122(2) Pages: 177-184
  • Peer Reviewed
• [Journal Article] Markov product of positive definite kernels ... (2011)
  • Author(s): Nobuaki Obata
  • Journal Title: Colloq. Math. Volume: 122 Pages: 177-184
  • Peer Reviewed
• [Presentation] Free Meixner laws and localization of quantum walks on spidernets (2014)
  • Author(s): Nobuaki Obata
  • Organizer: Non-commutative and infinite dimensional harmonic analysis : representation theory and probability theory
  • Place of Presentation: 九州大学
  • Year and Date: 2014-02-17
• [Presentation] Coupled oscillators on digraphs with self-adaptive dynamics (2014)
  • Author(s): Nobuaki Obata
  • Organizer: International Joint Research Week on Non-Commutative Stochastic Systems : Analysis, Modelling, and Applications
  • Place of Presentation: 東北大学
  • Year and Date: 2014-02-07
• [Presentation] Spectral analysis of digraphs and coupled oscillators (2013)
  • Author(s): Nobuaki Obata
  • Organizer: 3rd Tunisian-Japanese Conference : Geometric and Harmonic Analysis on Homogeneous Spaces and Applications
  • Place of Presentation: Yasmine Hammamet, Tunisia
  • Year and Date: 2013-12-19
• [Presentation] Spectral analysis of Manhattan street networks (2013)
  • Author(s): Nobuaki Obata
  • Organizer: Tunisian -Japanese Symposium on Science, Society and Technology (TJASSST 2013)
  • Place of Presentation: Yasmine Hammamet, Tunisia
  • Year and Date: 2013-11-15
• [Presentation] Spectra of Manhattan street digraphs for synchronization of coupled oscillators (2013)
  • Author(s): Nobuaki Obata
  • Organizer: NIMS Hot Topics Workshops on The 8th Jikji workshop : Infinite Dimensional Analysis and Quantum Probability
  • Place of Presentation: Daejeon, Korea
  • Year and Date: 2013-08-19
• [Presentation] Asymptotic spectral distributions of distance k-graphs of Cartesian product graphs (2012)
  • Author(s): Nobuaki Obata
  • Organizer: International Conference in Honour of Sixtieth Birthday of Rajendra Bhatia : Matrices and Operators Indian Institute of Science
  • Place of Presentation: Bangalore, India
  • Year and Date: 2012-12-29
• [Presentation] Quantum probability and asymptotic spectral analysis of growing graphs (2012)
  • Author(s): Nobuaki Obata
  • Organizer: International Conference on mathematical modeling and its aplications (ICMMA 2012)
  • Place of Presentation: Annamalainagar, India
  • Year and Date: 2012-12-23
• [Presentation] Asymptotic spectral analysis of growing graphs : A quantum probabilistic approach (2012)
  • Author(s): Nobuaki Obata
  • Organizer: International Conference on Frontiers of Statistics and its Applications
  • Place of Presentation: Puducherry, India
  • Year and Date: 2012-12-22
• [Presentation] Quantum probability and asymptotic spectral analysis of growing graphs (2012)
  • Author(s): Nobuaki Obata
  • Organizer: International Conference on Stochastic Modelling and Simulation (ICSMS2012)
  • Place of Presentation: Chennai, India
  • Year and Date: 2012-12-17
• [Presentation] Distance k-graphs of direct product graphs and their asymptotic spectral distributions (2012)
  • Author(s): Nobuaki Obata
  • Organizer: QP 33 "Quantum Probabilities and Related Fields" Centre International de Rencontres Mathematiques (CIRM)
  • Place of Presentation: Luminy, France
  • Year and Date: 2012-10-05
• [Presentation] Localization of quantum walks on spidernets (2012)
  • Author(s): Nobuaki Obata
  • Organizer: 15th Workshop on Non-Commutative Harmonic Analysis : Random Matrices, representation theory and free probability
  • Place of Presentation: with applications Mathematical Research and Conference Center (MRCC), Bedlewo, Poland
  • Year and Date: 2012-09-24
• [Presentation] Random walk, quantum walk, and quantum probability (2012)
  • Author(s): Nobuaki Obata
  • Organizer: 6th KMS Probability Workshop
  • Place of Presentation: Seoul, Korea
  • Year and Date: 2012-06-01
• [Presentation] Spectral analysis of digraphs (2012)
  • Author(s): Nobuaki Obata
  • Organizer: Workshop on Non-Commutative Harmonic Analysis with Applications to Real World Complex Phenomena
  • Place of Presentation: Hanyang University, Korea
  • Year and Date: 2012-01-16
• [Presentation] Random walk, quantum walk, and quantum probability (2012)
  • Author(s): Nobuaki Obata
  • Organizer: 6th Korean Mathenatical Society Probability Workshop
  • Place of Presentation: Seoul, Korea
  • Invited
• [Presentation] Localization of quantum walks on spidernets (2012)
  • Author(s): Nobuaki Obata
  • Organizer: 15th Workshop on Non-Commutative Harmonic Analysis: Random Matrices, representation theory and free probability, with applications
  • Place of Presentation: Bedlewo, Poland
  • Invited
• [Presentation] Distance k-graphs of direct product graphs and their asymptotic spectral distributions (2012)
  • Author(s): Nobuaki Obata
  • Organizer: QP 33: Quantum Probabilities and Related Fields
  • Place of Presentation: Luminy, France
• [Presentation] Asymptotic spectral analysis of growing graphs: A quantum probabilistic approach (2012)
  • Author(s): Nobuaki Obata
  • Organizer: International Conference on Frontiers of Statistics and its Applications
  • Place of Presentation: Puducherry, India
• [Presentation] Spectral analysis of digraphs (2012)
  • Author(s): Nobuaki Obata
  • Organizer: Workshop on Non-Commutative Harmonic Analysis ...(招待講演)
  • Place of Presentation: Seoul, Korea
• [Presentation] Quantum probabilistic spectral analysis of graphs (2011)
  • Author(s): Nobuaki Obata
  • Organizer: 数理解析研究所研究集会「独立性と従属性の数理 -- 代数と確率の出会い --」
  • Place of Presentation: 京都
  • Year and Date: 2011-12-19
• [Presentation] Quantum probabilistic spectral analysis of large graphs (2011)
  • Author(s): Nobuaki Obata
  • Organizer: The 2nd Tunisian--Japanese Conference : Geometric and Harmonic Analysis on Homogeneous Spaces and Applications
  • Place of Presentation: Sousse, Tunisia
  • Year and Date: 2011-12-14
• [Presentation] Spectral analysis of product graphs (2011)
  • Author(s): Nobuaki Obata
  • Organizer: 3rd Conference of Settat on Operator Algebras and Applications Marrakesh
  • Place of Presentation: Morocco
  • Year and Date: 2011-11-01
• [Presentation] Quantum probabilistic spectral analysis of networks and applications (2011)
  • Author(s): Nobuaki Obata
  • Organizer: International Conference on Stochastic Analysis and Applications Hammamet
  • Place of Presentation: Tunisia
  • Year and Date: 2011-10-11
• [Presentation] Spectral analysis of Manhattan products of digraphs (2011)
  • Author(s): Nobuaki Obata
  • Organizer: 14th Workshop : Non-commutative Harmonic Analysis
  • Place of Presentation: Bedlewo, Poland
  • Year and Date: 2011-09-25
• [Presentation] Spectral analysis of Manhattan products of ... (2011)
  • Author(s): Nobuaki Obata
  • Organizer: 14th Workshop: Non-commutative Harmonic Analysis(招待講演)
  • Place of Presentation: Bedlewo, Poland
• [Presentation] Quantum probabilistic spectral analysis of ... (2011)
  • Author(s): Nobuaki Obata
  • Organizer: Internat. Conf. Stochastic Anal. Appl.(招待講演)
  • Place of Presentation: Hammamet, Tunisia
• [Presentation] Spectral analysis of product graphs (2011)
  • Author(s): Nobuaki Obata
  • Organizer: 3rd Settat Conf. Operator Algebras and Appl.
  • Place of Presentation: Marrakesh, Morocco
• [Presentation] Quantum probabilistic spectral analysis of ... (2011)
  • Author(s): Nobuaki Obata
  • Organizer: 2nd Tunisian-Japanese Conf. Geo. Harm. Anal.
  • Place of Presentation: Sousse, Tunisia
• [Presentation] Free Meixner laws and localization of quantum walks on spidernets
  • Author(s): Nobuaki Obata
  • Organizer: Non-commutative and infinite dimensional harmonic analysis: representation theory and probability theory
  • Place of Presentation: 九州大学
• [Book] 確率モデル要論 (2012)
  • Author(s): 尾畑伸明
  • Publisher: 牧野書店
• [Remarks]
  • URL: http://www.math.is.tohoku.ac.jp
• [Remarks] 研究室ウェッブページ
  • URL: http://www.math.is.tohoku.ac.jp/~obata/