Google matrix analysis of directed networks Lecture 2 Klaus Frahm - PowerPoint PPT Presentation

Google matrix analysis of directed networks Lecture 2 Klaus Frahm Quantware MIPS Center Universit´ e Paul Sabatier Laboratoire de Physique Th´ eorique, UMR 5152, IRSAMC A. D. Chepelianskii, Y. H. Eom, L. Ermann, B. Georgeot, D. Shepelyansky Networks and data mining Luchon, June 27 - July 11, 2015

Contents Spectrum of Wikipedia . . . . . . . . . . . . . . . . . . . . 3 Eigenvectors of Wikipedia . . . . . . . . . . . . . . . . . . 4 Chirikov Standard map . . . . . . . . . . . . . . . . . . . . 9 Perron-Frobenius matrix for chaotic maps . . . . . . . . . . 10 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . 12 Complex density of states . . . . . . . . . . . . . . . . . . 13 Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . 14 Diffuson modes . . . . . . . . . . . . . . . . . . . . . . . . 19 Extrapolation of eigenvalues . . . . . . . . . . . . . . . . . 22 Strong chaos . . . . . . . . . . . . . . . . . . . . . . . . . 23 Separatrix map . . . . . . . . . . . . . . . . . . . . . . . . 26 Phase space localization . . . . . . . . . . . . . . . . . . . 27 References . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2

Spectrum of Wikipedia Wikipedia 2009 : N = 3282257 nodes, N ℓ = 71012307 network links. spectrum of S ∗ , N s = 21198 spectrum of S , N s = 515 n A = 6000 for both cases 3

Eigenvectors of Wikipedia left (right): PageRank (CheiRank) black: PageRank (CheiRank) at α = 0 . 85 grey: PageRank (CheiRank) at α = 1 − 10 − 8 red and green: first two core space eigenvectors blue and pink: two eigenvectors with large imaginary part in the eigenvalue 4

Detail study of 200 selected eigenvectors with eigenvalues “close” to the unit circle: 5

Power law decay of eigenvectors: | ψ i ( K i ) | ∼ K b K i ≥ 10 4 for i ϕ = arg( λ i ) 6

Inverse participation ratio of eigenvectors: j | ψ i ( j ) | 2 ) 2 / � j | ψ i ( j ) | 4 ξ IPR = ( � ϕ = arg( λ i ) 7

“Themes” of certain eigenvectors: math (function, geometry,surface, logic-circuit) England poetry Iceland aircraft Kuwait poetry Bangladesh football 0.5 biology song muscle-artery muscle-artery New Zeland DNA Austria Bible Poland muscle-artery music 0 -1 -0.5 0 0.5 1 Australia Canada protein Brazil China RNA skin war rail 0 Texas-Dallas-Houston -0.82 -0.8 -0.78 -0.74 -0.72 Gaafu Alif Atoll -0.76 Quantum Leap Language Switzerland Australia Australia England mathematics 0 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 8

Chirikov Standard map 0.5 p n +1 = p n + k 0.4 2 π sin(2 π x n ) 0.3 p 0.2 x n +1 = x n + p n +1 0.1 k=0.7 0 0 0.2 0.4 0.6 0.8 1 x and p are taken modulo 1 x and the symmetry 0.5 ( x, p ) → (1 − x, 1 − p ) 0.4 allows to restrict: 0.3 p 0.2 x ∈ [0 , 1] and p ∈ [0 , 0 . 5] . 0.1 k=0.971635406 Transition to “global” chaos at 0 0 0.2 0.4 0.6 0.8 1 k c = 0 . 971635406 . x 9

Perron-Frobenius matrix for chaotic maps A new variant of the Ulam Method to construct the Perron-Frobenius matrix for the case of a mixed phase space: • Subdivide x space in M cells and p space in M/ 2 cells with M being an (even) integer number. • Iterate (for a very long time: t ∼ 10 11 − 10 12 ) a classical trajectory and attribute a new number to each new cell which is entered. At the same time count the number of transitions from cell i to cell j ( ⇒ n ji ). • Calculate the N × N matrix n ji G ji = � l n li of dimension N ≈ M 2 / 4 and which is a (sparse) Perron Frobenius operator, i. e.: G ji ≥ 0 , � j G ji = 1 , G ji sparse. 10

M = 10 , t = 10 6 ⇒ N = 35 0.5 0.4 3 0.3 4 p 0.2 6 2 0.1 0 1 5 0 0 0.2 0.4 0.6 0.8 1 x 14 12 10 8 6 4 2 0 0 1 2 3 4 5 6 distribution of number of non-zero density plot of matrix elements matrix elements per column (blue=min=0, green=medium, red=max) 11

Eigenvalues for M = 10 , t = 10 6 and N = 35 1 0.5 0 -0.5 λ Phase space representation of the -1 -1 -0.5 0 0.5 1 eigenvector for λ 0 = 1 . for M = 280 , t = 10 12 and N = 16609 1 0.01 0.5 0 0 -0.5 λ -0.01 λ -1 -1 -0.5 0 0.5 1 0.97 0.98 0.99 1 12

Complex density of states M=280 1.2 M=200 M=140 1 M=100 0.8 ρ ( λ ) 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 | λ | 13

Eigenvectors λ 0 = 1 , M = 25 , N = 177 λ 0 = 1 , M = 35 , N = 332 λ 0 = 1 , M = 50 , N = 641 λ 0 = 1 , M = 70 , N = 1189 14

λ 0 = 1 M = 140 N = 4417 λ 0 = 1 M = 280 N = 16609 15

complex density of states: 0.3 M=280 M=400 M=560 M=800 0.2 M=1120 M=1600 ρ ( λ ) 0.1 0 0.6 0.8 1 | λ | 16

λ 0 = 1 , M = 1600 , N = 494964 , n A = 3000 17

λ 1 = 0 . 99980431 M = 800 N = 127282 n A = 2000 λ 2 = 0 . 99878108 M = 800 N = 127282 n A = 2000 18

Diffuson modes 0.03 γ 1 j 2 γ j = − 2 ln( | λ j | ) 0.02 γ j γ j ≈ γ 1 j 2 0.01 for j ≤ 5 . 0 0 5 10 15 20 j What about eigenvectors for complex or real negative λ j ? 19

λ 6 = − 0 . 49699831 + i 0 . 86089756 ≈ | λ 6 | e i 2 π/ 3 M = 800 N = 127282 n A = 2000 λ 19 = − 0 . 71213331 + i 0 . 67961609 ≈ | λ 19 | e i 2 π (3 / 8) M = 800 N = 127282 n A = 2000 20

λ 8 = 0 . 00024596 + i 0 . 99239222 ≈ | λ 8 | e i 2 π/ 4 M = 800 N = 127282 n A = 2000 λ 13 = 0 . 30580631 + i 0 . 94120900 ≈ | λ 13 | e i 2 π/ 5 M = 800 N = 127282 n A = 2000 21

Extrapolation of eigenvalues γ 1 ( M ) in the limit M → ∞ : 1 + C f ( M ) = D M 0.1 f(M) 1 + B M 0.01 M γ 1 (M) D = 0 . 245 2.36 M -1.30 0.001 B = 13 . 1 10 -4 100 1000 C = 258 M γ 6 ( M ) in the limit M → ∞ : 0.1 389 M -1.55 γ 6 (M) 0.01 200 400 800 1600 M γ 6 ( M ) ≈ 389 M − 1 . 55 for M ≥ 400 . 22

Strong chaos at k = 7 , M = 140 , t = 10 11 and N = 9800 0.3 d+e M -1 1 a (1+b M -1 ) c 0.2 γ 1 (M) 0.5 0.1 γ class. =0.0866 0 0 0 0.005 0.01 -0.5 M -1 λ a ≈ 0 . 0857 ± 0 . 0036 , d ≈ 0 . 0994 -1 -1 -0.5 0 0.5 1 M →∞ γ 1 ( M ) > 0 lim 23

λ 0 = 1 k = 7 M = 800 N = 319978 n A = 1500 λ 1 = 0 . 93874817 k = 7 M = 800 N = 319978 n A = 1500 24

λ 2 = − 0 . 49264273 + i 0 . 78912368 k = 7 M = 800 N = 319978 n A = 1500 λ 8 = 0 . 87305253 k = 7 M = 800 N = 319978 n A = 1500 25

Separatrix map p = p + sin(2 πx ) ¯ x = x + Λ ¯ 2 π ln( | ¯ p | ) (mod 1) Λ = Λ c = 3 . 1819316 26

Phase space localization 27

References 1. D. L. Shepelyansky Fractal Weyl law for quantum fractal eigenstates , Phys. Rev. E 77 , p.015202(R) (2008). 2. L. Ermann and D. L. Shepelyansky, Ulam method and fractal Weyl law for Perron-Frobenius operators , Eur. Phys. J. B 75 , 299 (2010). 3. K. M. Frahm and D. L. Shepelyansky, Ulam method for the Chirikov standard map , Eur. Phys. J. B 76 , 57 (2010). 4. L. Ermann, K. M. Frahm and D. L. Shepelyansky, Spectral properties of Google matrix of Wikipedia and other networks , Eur. Phys. J. B 86 , 193 (2013). 5. K. M. Frahm and D. L. Shepelyansky, Poincar´ e recurrences and Ulam method for the Chirikov standard map , Eur. Phys. J. B 86 , 322 (2013). 28