Asymptotic Spectral Analysis of Growing Graphs Nobuaki Obata - PowerPoint PPT Presentation

Asymptotic Spectral Analysis of Growing Graphs Nobuaki Obata Graduate School of Information Sciences Tohoku University www.math.is.tohoku.ac.jp/˜obata SJTU, Shanghai, China, 2018.11.15–18 Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 1 / 80

Introducing myself... Tianjin Sendai Tohoku University 1 — The 3rd oldest national University of Japan, founded in 1907. Graduate School of Information Sciences (GSIS) 2 — One of the 17 Graduate Schools, founded in 1993. Nobuaki Obata — Serving as Professor since 2001. 3 Before then I was a member of Department of Mathematics in Nagoya University. Major research interests — Spectral analysis of graphs, Random graphs, Quantum 4 probability, Quantum white noise analysis, and any topics related to network science. Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 2 / 80

Main Theme: Asymptotic Spectral Analysis of Growing Graphs ▶ Spectral analysis of graphs ( ) � dx = f x dx ( ) ( ) G = V, E [ ] A ���� A xy ▶ Growing graphs Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 3 / 80

My Motivations and Backgrounds (1) Statistics in large scale discrete systems A. M. Vershik’s asymptotic combinatorics (1970s–) 1 [1] Asymptotic combinatorics and algebraic analysis (ICM 1994) ... the study of asymptotic problems in combinatorics is stimulated enormously by taking into account the various approaches from different branches of mathematics. ... The main question in this context is: What kind of limit behavior can have a combinatorial object when it “grows” ? [2] Between “very large” and “infinite” (Bedlewo 2012) [3] Takagi lecture of Mathematical Society of Japan (Tohoku University, 2015) [4] see also A. Hora: The limit shape problem for emsembles of Young diagrams, Springer 2017. Complex networks — modelling real world large networks 2 [1] A.-L. Barab´ asi and R. Albert (1999) — scale free networks [2] D. J. Watts and S. H. Strogatz (1998) — small world networks [3] F. Chung and L. Lu (2006), R. Durrett (2007), L. Lovasz (2012). Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 4 / 80

My Motivations and Backgrounds (2) Quantum probability = Noncommutative Probability = Algebraic Probability J. von Neumann: Mathematische Grundlagen der Quantenmechanik (1932) 1 Mathematical theory for the probabilistic interpretation in quantum mechanics in terms of operators on Hilbert spaces. The term quantum probability was introduced actively by L. Accardi (Roma) around 2 1978. R. Hudson and K. R. Parthasarathy (1984) initiated quantum Ito calculus. 3 P.-A. Meyer: Quantum Probability for Probabilists , LNM 1538 (1993). 4 N. Obata: Quantum probability + graph theory and network science since 1998. 5 [1] A. Hora and N. Obata: Quantum Probability and Spectral Analysis of Graphs , Springer, 2007. [2] N. Obata: Spectral Analysis of Growing Graphs. A Quantum Probability Point of View , Springer, 2017. Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 5 / 80

Plan Spectral Distributions of Graphs 1 Method of Quantum Decomposition 2 Asymptotic Spectral Analysis of Growing Regular Graphs 3 Graph Products and Concepts of Independence 4 Summary and Perspectives 5 ▶ Main Reference N. Obata: Spectral Analysis of Growing Graphs — A Quantum Probability Point of View , Springer, 2017. Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 6 / 80

Spectral Distributions of Graphs 1. Spectral Distributions of Graphs [Chapters 1–3] Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 7 / 80

Spectral Distributions of Graphs 1.1. Quantum Probability — Algebraic Probability Spaces Definition A pair ( A , φ ) is called an algebraic probability space if A is a unital ∗ -algebra over C and φ a state on it, i.e., (i) φ : A → C is a linear function; (ii) positive, i.e., φ ( a ∗ a ) ≥ 0 ; (iii) normalized, i.e., φ (1 A ) = 1 . Definition Each a ∈ A is called an (algebraic) random variable . It is called real if a = a ∗ . ▶ (Ω , F , P ) : classical (Kolmogorovian) probability space ∩ A = L ∞− (Ω , F , P ) = L p (Ω , F , P ) 1 ≤ p< ∞ = { X : Ω → C ; E[ | X | m ] < ∞ for all m ≥ 1 } φ ( X ) = E[ X ] , X ∈ A . Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 8 / 80

Spectral Distributions of Graphs 1.1. Quantum Probability — Statistics Definition (1) For a random variable a ∈ A , its mixed moments are defined by φ ( a ϵ m · · · a ϵ 2 a ϵ 1 ) , ϵ 1 , ϵ 2 . . . , ϵ m ∈ { 1 , ∗} . (2) For a real random variable a = a ∗ ∈ A the mixed moments are reduced to the moment sequence : φ ( a m ) , m = 1 , 2 , . . . . Definition (1) Two algebraic random variables a in ( A , φ ) and b in ( B , ψ ) are called stochastically equivalent if their all mixed moments coincide: φ ( a ϵ m · · · a ϵ 2 a ϵ 1 ) = ψ ( b ϵ m · · · b ϵ 2 b ϵ 1 ) . (2) Two real random variables a = a ∗ in ( A , φ ) and b = b ∗ in ( B , ψ ) are stochastically equivalent if φ ( a m ) = ψ ( b m ) , m = 1 , 2 , . . . . Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 9 / 80

Spectral Distributions of Graphs 1.1. Quantum Probability — Spectral Distributions Theorem (spectral distribution) For a real random variable a = a ∗ ∈ A there exists a probability measure µ on R = ( −∞ , + ∞ ) such that ∫ + ∞ φ ( a m ) = x m µ ( dx ) ≡ M m ( µ ) , m = 1 , 2 , . . . . −∞ This µ is called the spectral distribution of a in the state φ . Existence proof is by Hamburger’s theorem using Hanckel determinants. 1 µ is not uniquely determined in general (determinate moment problem). 2 µ is unique, for example, if 3 ∞ ∑ 1 − M 2 m = + ∞ (Carleman’s moment test) 2 m m =1 = ⇒ Details omitted (see Chapters 4-5) Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 10 / 80

Spectral Distributions of Graphs 1.1. Quantum Probability vs Classical Probability Classical Probability Quantum Probability probability space (Ω , F , P ) ( A , φ ) a = a ∗ ∈ A random variable X : Ω → R ∫ expectation E[ X ] = X ( ω ) P ( dω ) φ ( a ) Ω distribution µ X (( −∞ , x ]) = P ( X ≤ x ) NA E[ X m ] φ ( a m ) moments ∫ + ∞ ∫ + ∞ E[ X m ] = x m µ X ( dx ) φ ( a m ) = x m µ a ( dx ) −∞ −∞ Definition (algebraic realization) An algebraic random variable a = a ∗ in an algebraic probability space ( A , φ ) is called an algebraic realization of a classical random variable X if their moments coincide: φ ( a m ) = E[ X m ] , m = 1 , 2 , . . . . Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 11 / 80

Spectral Distributions of Graphs 1.2. Matrix Algebras — States on M ( n, C ) Equipped with the usual matrix operations, A = M ( n, C ) = { a = [ a ij ] ; a ij ∈ C } becomes a unital ∗ -algebra over C . (i) the normalized trace: n φ ( a ) = 1 n Tr ( a ) = 1 ∑ a ii , a = [ a ij ] . n i =1 (ii) a vector state: ξ ∈ C n , φ ( a ) = ⟨ ξ, aξ ⟩ , ∥ ξ ∥ = 1 . Lemma (exercise) A general form of a state on M ( n, C ) is given by φ ( a ) = Tr ( ρa ) , where ρ is a density matrix, i.e., ρ = ρ ∗ ≥ 0 and Tr ( ρ ) = 1 . Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 12 / 80

Spectral Distributions of Graphs 1.2. Matrix Algebras — A Model of Coin Toss Traditional Model for Coin-toss A random variable X on a probability space (Ω , F , P ) satisfying the property: P ( X = +1) = P ( X = − 1) = 1 2 More essential is the probability distribution of X : µ X = 1 2 δ − 1 + 1 2 δ +1 Lemma (Moment sequence) Let X be the coin toss defined as above. Then we have  ∫ + ∞  1 , if m is even , E[ X m ] = M m ( µ X ) = x m µ X ( dx ) = 0 , otherwise .  −∞ Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 13 / 80

Spectral Distributions of Graphs 1.2. Matrix Algebras — A Model of Coin Toss (cont) Set 1 [ ] [ ] [ ] 0 1 0 1 A = , e 0 = , e 1 = . 1 0 1 0 Define an algebraic probability space ( A , φ ) by 2 A = ∗ -algebra generated by A ; φ ( a ) = ⟨ e 0 , ae 0 ⟩ , a ∈ A . It is straightforward to see that 3   1 , if m is even , φ ( A m ) = ⟨ e 0 , A m e 0 ⟩ = 0 , otherwise ,  Thus, φ ( A m ) = E[ X m ] , m = 1 , 2 , . . . . Namely, A is an algebraic realization of the coin toss X . 4 Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 14 / 80