Quantum probabilistic analysis of dynamics on complex networks

2016 Fiscal Year Final Research Report

• Project/Area Number: 26610018
• 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): KONNO Norio 横浜国立大学, 大学院工学研究院, 教授 (80205575) ; HASEGAWA Takehisa 茨城大学, 理学部, 准教授 (10528425)
• Project Period (FY): 2014-04-01 – 2017-03-31
• Keywords: スペクトル解析 / 量子確率論 / 複雑ネットワーク / グラフのスペクトル / 振動子系 / 量子ウォーク / 有向グラフ / 直交多項式

Outline of Final Research Achievements

From quantum probability point of view we aimed at developing mathematical approach to complex networks. We obtained formulas for spectral distributions of certain graph products. These are applied to counting walks on two-dimensional restricted lattices and the density functiones are expressed in terms of elliptic integrals. We obtained the long-time asymptotics of quantum walks on the semi-infinite line by means of Chebyshev polynomials and examined the experimental data. We obtain the eigenvalues of the adjacency matrices and Laplacians of the Manhattan product of directed paths and, with the aid of numerical analysis, examined the behaviour of coupled oscillators. As for multi-variable polynomials, we obtained Gauss-Poisson distribution as a limit spectral distribution of strong regular graphs by means of two-variable Krautchuk polynomials.

Free Research Field

量子確率論・スペクトル解析