Google Matrix, dynamical attractors and Ulam networks Dima Shepelyansky (CNRS, Toulouse)

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1 Google Matrix, dynamical attractors and Ulam networks Dima Shepelyansky (CNRS, Toulouse) wwwquantwareups-tlsefr/dima based on: OGiraud, BGeorgeot, DLS (CNRS, Toulouse) => PRE 8, 267 (29) DLS, OVZhirov (CNRS, Toulouse & BINP, Novosibirsk) => arxiv:95462 WWW of sites Information search engines PageRank algorithm Network models Dynamical attractors, Ulam networks Fractal Weyl law S Brin and L Page, Computer Networks and ISDN Systems,7 (998) (Quantware group, CNRS, Toulouse) ICTP, Trieste 24/9/29 / 2

2 How Google works PageRank Algorithm Ranking pages {,,N} according to their importance Method: The importance of a page i depends on the importance of the pages j pointing on it If a page has many outgoing links the importance it transmits is proportional to the number of pages it points to The Google Matrix: G = αs + ( α)e/n here E is such a matrix that E ij = With a certain (stochastic) matrix S p = Sp (Quantware group, CNRS, Toulouse) ICTP, Trieste 24/9/29 2 / 2

3 How Google works Directed networks Weighted adjacency matrix 2 S = 2 For a directed network with N nodes the adjacency matrix A is defined as A ij = if there is a link from node j to node i and A ij = otherwise The weighted adjacency matrix is S ij = A ij / k A kj In addition the elements of columns with only zeros elements are replaced by /N (Quantware group, CNRS, Toulouse) ICTP, Trieste 24/9/29 / 2

4 How Google works Computation of PageRank p = Sp p= stationary vector of S: can be computed by iteration of S To remove convergence problems: Replace columns of (dangling nodes) by N : In our example, S = To remove degeneracies of the eigenvalue, replace S by G = αs + ( α) E ; Gp = λp => Perron-Frobenius operator N α models a random surfer with a random jump after approximately 6 clicks (usually α = 85); PageRank vector => p at λ = ( j p j = ) (Quantware group, CNRS, Toulouse) ICTP, Trieste 24/9/29 4 / 2

5 Models of real networks Real networks are characterized by: small world property: average distance between 2 nodes log N scale-free property: distribution of the number of outgoing or incoming links P(k) k γ Can be explained by a twofold mechanism: Constant growth: new nodes appear regularly and are attached to the network Preferential attachment: nodes are preferentially linked to already highly connected vertices PageRank vector for large WWW: p j /j β, wher j is the ordered index number of nodes N n with PageRank p scales as N n /p ν with numerical values ν = + /β 2 and β 9 (Quantware group, CNRS, Toulouse) ICTP, Trieste 24/9/29 5 / 2

6 Albert-Barabasi (AB) model Weight of a link chosen to be Π i = k i + j (k j + ), with k i being number of incoming+outgoing links Procedure from m nodes: starting from m nodes, at each step m links are added to the existing network with probability p (preferential attachment, start nodes are chosen randomly) or m links are rewired with probability q or a new node with m links is added with probability p q End vertex i always chosen with probability Π i We fix m = 5, p = 2, q = (scale-free) and q = 7 (exponential regimes) R Albert and A-L Barabási, Phys Rev Lett 85, 524 (2) (Quantware group, CNRS, Toulouse) ICTP, Trieste 24/9/29 6 / 2

7 Eigenvalues of Google Matrix for AB model Distribution of eigenvalues λ i of Google matrices in the complex plane Color is proportional to the PAR ξ of the associated eigenvector ψ i Top panel: AB model with q = for N = 2 4 for N r = 5 random realizations (see text), ξ varies from ξ = 2 (blue) to ξ = 656 (red); middle panel: same with q = 7, ξ varies from ξ = 69 (red) to ξ = 584 (purple); bottom panel: data for a university network (Liverpool J Moores Univ - LJMU) with N = 578, here in order to get statistically significant data the WWW network was randomized and data correspond to N r = 5 random realizations, ξ varies from ξ = 7 (blue) to ξ = 77 (red) (participation ratio PAR => ξ = ( P j ψ i(j) 2 ) 2 / P j ψ i(j) 4 ; Gψ i = λ i ψ i ) (Quantware group, CNRS, Toulouse) ICTP, Trieste 24/9/29 7 / 2

8 Density of states and participation ratio 4 ξ W(γ) W(γ) ξ γ γ Normalized density of states W (top panel) and PAR (bottom panel) as a function of γ Data for AB model with q = are shown by full curves with from bottom to top N = 2 (N r = ) (black), 2 (N r = 5) (red), 2 2 (N r = 2) (green), 2 (N r = ) (blue), 2 4 (N r = 5) (violet) Symbols give the PageRank value of ξ in the same order: circle, square, diamond, triangle down and triangle up All curves coincide on the top panel Dashed curves show the data from the WWW (LJMU network, parameters of previous Fig) Here λ = exp( γ/2) Same for AB model at q = 7 (Quantware group, CNRS, Toulouse) ICTP, Trieste 24/9/29 8 / 2

9 Dependence on matrix size N log ξ log N Dependence of ξ on matrix size N for AB model at q = (triangles), q = 7 (circles), and for WWW data without randomization (squares) Full symbols are for PageRank ξ values, empty symbols are for eigenvectors with < γ < 4 (AB model) or for the eigenvectors with highest ξ and γ < (WWW data) For AB model N r is as before and N r = 5 for N > 2 4 (statistical error bars are smaller than symbol size) Dotted blue lines give linear fits of WWW data, with slopes respectively and 5 Upper dashed line indicates the slope Logarithms are decimal (Quantware group, CNRS, Toulouse) ICTP, Trieste 24/9/29 9 / 2

10 Shapes of eigenvectors log ψ i (j) - -2 log P c (p j ) log p j log j Dependence of eigenvectors ψ i (j) of AB model on index j ordered in decreasing PageRank values p j (with normalisation P j ψ i(j) 2 = and P j p j = ) Full smooth curves are PageRank vectors for N = 2 4, dashed smooth curves for N = 2 9 Non-smooth curves are eigenvectors (N = 2 4 ) within < γ < 4 with Ψ i (j) 2 averaged in this interval States are averaged over N r = 5 random networks Black is for q =, red/grey for q = 7 Inset: cumulative distribution P c(p j ) normalized by P c() = N for AB model (N = 2 8 and N r = 5) at q = (full black) and q = 7 (dashed red/grey), and for LJMU non-randomized data (full red/grey) Dashed straight line indicates slope ν = Logarithms are decimal (Quantware group, CNRS, Toulouse) ICTP, Trieste 24/9/29 / 2

11 Ulam networks Ulam conjecture (method) for discrete approximant of Perron-Frobenius operator of dynamical systems SMUlam, A Collection of mathematical problems, Interscience, 8, 7 NY (96) A rigorous prove for hyperbolic maps: T-YLi JApprox Theory 7, 77 (976) Related works: Z Kovacs and T Tel, Phys Rev A 4, 464 (989) MBlank, GKeller, and CLiverani, Nonlinearity 5, 95 (22) DTerhesiu and GFroyland, Nonlinearity 2, 95 (28) Links to Markov chains: Contre-example: Hamiltonian systems with invariant curves, eg the Chirikov standard map: noise, induced by coarse-graining, destroys the KAM curves and gives homogeneous ergodic eigenvector at λ = thanks to Arkady Pikovsky for a remark about the Ulam method (Quantware group, CNRS, Toulouse) ICTP, Trieste 24/9/29 / 2

12 Google matrix of dynamical attractors Weak point of AB model => large gap, no sensitivity to α PageRank p j for the Google matrix generated by the Chirikov typical map at T =, k = 22, η = 99 (set T, top row) and T = 2, k =, η = 97 (set T 2, bottom row) with α =, 95, 85 (left to right) The phase space region x < 2π; π p < π is divided on N = 6 5 cells Chirikov typical map (969) with dissipation p = ηp + k sin(x + θ t), x = x + p θ t = θ t+t are random phases periodically repeated after T iterations, chaos border k c 25/T /2, Kolmogorov-Sinai entropy h 29k 2/ ; grid of N = N x N p cells with N c 4 trajectories which generates links (transition probabilities) from one cell to another; effective noise of cell size; maximum N = 225; 44 6 (Quantware group, CNRS, Toulouse) ICTP, Trieste 24/9/29 2 / 2

13 Bifurcation diagram <= Bifurcation diagram showing values of p vs map parameter k for the set T The values of p, obtained from trajectories with initial random positions in the phase space region, are shown for integer moments of time < t/t (left) and 4 < t/t 4 + (right) <= Same for set T 2 (Quantware group, CNRS, Toulouse) ICTP, Trieste 24/9/29 / 2

14 Distribution of links log Pin,out log κ 2 log κ Differential distribution of number of nodes with ingoing P in (κ) and outgoing P out(κ) links κ for sets T (left) and T 2 (right) The straight dashed lines give the algebraic fit P(κ) κ µ with the exponent µ = 86, (T, T 2) for ingoing and µ = 9, 46 (T, T 2) outgoing links Here N = 44 6 and P(κ) gives a number of nodes at a given integer number of links κ for this matrix size Blue point at κ = shows that in the whole matrix there is a significant number of nodes with zero ingoing links Typical number of nodes κ exp(ht) (Quantware group, CNRS, Toulouse) ICTP, Trieste 24/9/29 4 / 2

15 PageRank distribution -2 (a) (b) (c) -4 log pj -2 (d) (e) (f) -4 (j/) log j log j Differential distribution of number of nodes with PageRank distribution p j for N = 4, 9 4, 6 5 and 44 6 curves, the dashed straight lines show fits p j /j β with β: 48 (b), 88 (e), 6 (f) Dashed lines in panels (a),(d) show an exponential Boltzmann decay (see text, lines are shifted in j for clarity) In panels (a),(d) the curves at large N become superimposed Panel order as in color Fig above (Quantware group, CNRS, Toulouse) ICTP, Trieste 24/9/29 5 / 2

16 Properties of eigenvalues and eigenvectors log( λ ) ξ (a) (N/ 4 ) -6 2 (b) (c) (d) γ (a) Dependence of gap λ on Google matrix size N for few eigenstates with λ most close to, set T, α = ; (b) dependence of PAR ξ on γ = 2 ln λ for N=25, 5625, 8, 4, 44 for set T, α = ; (c) plane of eigenvalues λ for set T with their PAR ξ values shown by grayness (black/blue for minimal ξ 4, gray/light magenta for maximal ξ ; here α =, N = 44 4 ); (d) same as (c) but for set T 2 (Quantware group, CNRS, Toulouse) ICTP, Trieste 24/9/29 6 / 2

17 Fractal Weyl law invented for open quantum systems: the number of Gamow eigenstates N γ, that have escape rates γ in a finite bandwidth γ γ b, scales as N γ (d ) where d is a fractal dimension of a strange repeller formed by obits non-escaping in future References: JSjostrand, Duke Math J 6, (99) MZworski, Not Am Math Soc 46, 9 (999) WTLu, SSridhar and MZworski, Phys Rev Lett 9, 54 (2) SNonnenmacher and MZworski, Commun Math Phys 269, (27) Quantum Chirikov standard map with absorption FBorgonovi, IGuarneri, DLS, Phys Rev A 4, 457 (99) DLS, Phys Rev E 77, 522(R) (28) Perron-Frobenius operators? (Quantware group, CNRS, Toulouse) ICTP, Trieste 24/9/29 7 / 2

18 Fractal Weyl law for the Chirikov standard map Leonardo Ermann (CNRS, Toulouse) => poster N= x, k=7, a=2 4 2 Im(λ) Re(λ) λ = γ c = 26, K = 7 (Quantware group, CNRS, Toulouse) ICTP, Trieste 24/9/29 8 / 2

19 Fractal Weyl law and distribution over γ dw/dγ 5 log Nγ 5 4 log N γ fractal Weyl: N γ N ν ; ν = d = γ c /(Th), γ c = T ln η theory: ν = 88; 72 numerics: ν = 85; 6 almost all states have λ = Probability distribution dw(γ)/dγ for set T, α = at N = 25 ( ), 4 (+), 44 4 (dots); W(γ) is normalized by the number of states N γ = 55N 85 with γ < 6 Inset: dependence of number of states N γ with γ < γ b on N for sets T (circles, γ b = 6) and T 2 (triangles, γ b = ); dashed lines show the fit N γ = AN ν with A = 55, ν = 85 and A = 97, ν = 6 respectively (Quantware group, CNRS, Toulouse) ICTP, Trieste 24/9/29 9 / 2

20 Delocalization transition in α and k log ξ log ξ k α delocalization of PageRank with α and k => destruction of Google search efficiency α c 95; 85 for T; T2; α c γ c Dependence of PageRank ξ on α for set T at N = 5625 (dotted magenta), 44 4 (dotted red), 9 4 (dashed red), 64 5 (full red) and for T 2 at N = 44 4 (dotted blue), 9 4 (dashed blue), 64 5 (full blue) Inset shows dependence of ξ on k for set T at α = 99 with N = 44 4 (dotted red), 9 4 (dashed red), 6 5 (full red) (Quantware group, CNRS, Toulouse) ICTP, Trieste 24/9/29 2 / 2

21 Summary T: α = 99, N = 6 5 k = 22 (localized, left) k = 6 (delocalized, right) popular web sites are like attractors delocalization => transition to strange attractor links between directed networks and dynamical systems delocalization transition can put in danger Google search interesting new physics of Google Matrix (Quantware group, CNRS, Toulouse) ICTP, Trieste 24/9/29 2 / 2