Time-Series Based Prediction of Dynamical Systems and Complex. Networks

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1 Time-Series Based Prediction of Dynamical Systems and Complex Collaborators Networks Ying-Cheng Lai ECEE Arizona State University Dr. Wenxu Wang, ECEE, ASU Mr. Ryan Yang, ECEE, ASU Mr. Riqi Su, ECEE, ASU Prof. Celso Grebogi, Univ. of Aberdeen Dr. Vassilios Kovanis, AFRL, Wright-Patterson AFB

2 Problem 1: Can catastrophic events in dynamical systems be predicted in advance? Early Warning? A related problem: Can future behaviors of time-varying dynamical systems be forecasted?

3 Problem 2: Reverse-engineering of complex networks network Measured Time Series Assumption: all nodes are externally accessible Full network topology?

4 Problem 3: Detecting hidden nodes No information is available from the black node. How can we ascertain its existence and its location in the network?

5 Basic idea (1 Dynamical system: dx/dt = F(x, x! R m Goal: to determine F(x from measured time series x(t! Power-series expansion of jth component of vector field F(x n n " l 1 =0 l 2 =0 n " l m =0 [F(x] j = " l... (a j l1 l 2...l m x 1 l 1 x 2 l 2...x m m x k kth component of x; Highest-order power: n (a j l1 l 2...l m - coefficients to be estimated from time series - (1+n m coefficients altogether If F(x contains only a few power-series terms, most of the coefficients will be zero.

6 Basic idea (2 Concrete example: m = 3 (phase-space dimension: (x,y,z n = 3 (highest order in power-series expansion total (1 + n m = ( = 64 unknown coefficients [F(x] 1 = (a 1 0,0,0 x 0 y 0 z 0 + (a 1 1,0,0 x 1 y 0 z (a 1 3,3,3 x 3 y 3 z 3! Coefficient vector a 1 = " (a 1 0,0,0 (a 1 1,0,0... (a 1 3,3,3-64 '1 Measurement vector g(t = [x(t 0 y(t 0 z(t 0, x(t 1 y(t 0 z(t 0,..., x(t 3 y(t 3 z(t 3 ] So [F(x(t] 1 = g(t a 1 1 ' 64

7 Basic idea (3 Suppose x(t is available at times t 0, t 1,t 2,...,t 10 (11 vector data points dx dt (t 1 = [F(x(t 1 ] 1 = g(t 1 a 1 dx dt (t 2 = [F(x(t 2 ] 1 = g(t 2 a 1... dx dt (t 10 = [F(x(t 10 ] 1 = g(t 10 a 1 Derivative vector dx =! " (dx/dt(t 1 (dx/dt(t 2... (dx/dt(t 10 10'1 ; Measurement matrix G =! " g(t 1 g(t 2! g(t 10 10'64 We finally have dx = G a 1 or dx 10'1 = G 10'64 (a 1 64'1

8 Basic idea (4 dx = G a 1 or dx 10!1 = G 10!64 (a 1 64!1 Reminder: a 1 is the coefficient vector for the first dynamical variable x. To obtain [F(x] 2, we expand [F(x] 2 = (a 2 0,0,0 x 0 y 0 z 0 + (a 2 1,0,0 x 1 y 0 z (a 2 3,3,3 x 3 y 3 z 3 with a 2, the coefficient vector for the second dynamical variable y. We have dy = G a 2 or dy 10!1 = G 10!64 (a 2 64!1 where dy = " (dy/dt(t 1 (dy/dt(t 2... (dy/dt(t 10 ' ' ' ' 10!1 Note: the measurement matrix G is the same.. Similar expressions can be obtained for all components of the velocity field.

9 Compressive sensing (1 Look at dx = G a 1 or dx 10!1 = G 10!64 (a 1 64!1 Note that a 1 is sparse - Compressive sensing! Data/Image compression:! : Random projection (not full rank x - sparse vector to be recovered Goal of compressive sensing: Find a vector x with minimum number of entries subject to the constraint y =! x

10 Compressive Sensing (2 Find a vector x with minimum number of entries subject to the constraint y =! x: l 1 " norm Why l 1! norm? - Simple example in three dimensions E. Candes, J. Romberg, and T. Tao, IEEE Trans. Information Theory 52, 489 (2006, Comm. Pure. Appl. Math. 59, 1207 (2006; D. Donoho, IEEE Trans. Information Theory 52, 1289 (2006; Special review: IEEE Signal Process. Mag. 24, 2008

11 Predicting catastrophe (1 Henon map: (x n+1, y n+1 = (1- ax 2 n + y n, bx n Say the system operates at parameter values: a = 1.2 and b = 0.3. There is a chaotic attractor. Can we assess if a catastrophic bifurcation (e.g., crisis is imminent based on a limited set of measurements? Step 1: Predicting system equations Distribution of predicted values of ten power-series coefficients: constant, y, y 2, y 3, x, xy, xy 2, x 2, x 2 y, x 3 of data points used: 8

12 Predicting catastrophe (2 Step 2: Performing numerical bifurcation analysis Current operation point Boundary Crisis

13 Predicting catastrophe (3 Examples of predicting continuous-time dynamical systems Classical Lorenz system dx/dt = 10y - 10x dy/dt = x(a - z - y dz/dt = xy - (2/3z Classical Rossler system dx/dt = -y - z dy/dt = x + 0.2y dz/dt = z(x - a of data points used: 18

14 Performance analysis Standard map Lorenz system n m! of measurements n nz! of non-zero coefficients; n z! of zero coefficients (n nz + n z! total of coefficients to be determined n t! minimum of measurements required for accurate prediction

15 Effect of noise Henon map Standard map n m = 8 (n nz + n z =16 n m =10 (n nz + n z = 20 W.-X. Wang, R. Yang, Y.-C. Lai, V. Kovanis, and C. Grebogi, Physical Review Letters 106, (2011.

16 Predicting future attractors of time-varying dynamical systems (1 Dynamical system: dx/dt = F[x,p(t], p(t - parameters varying slowly with time T M " measurement time period; x! R m x(t - available in time interval: t M -T M! t! t M Goal: to determine both F[x,p(t] and p(t from available time series x(t so that the nature of the attractor for t > t M can be assessed. Power-series expansion n n n l [F(x] j =...(! j l1 l 2...l m x 1 l 1 x 2 l 2...x m m (! j w t w n v l 1 =0 l 2 =0 l m =0 = (c j x l 1 l l1,...,l m ;w 1 x 2 l 2...x m m t w CS framework l 1,...,l m =0 w=0 v w=0

17 Predicting future attractors of time-varying dynamical systems (2 Formulated as a CS problem:

18 Predicting future attractors of time-varying dynamical systems (3 z z (b time (a time Measurements Actual time series Predicted time series k 1 (t = 0 k 2 (t = 0 k 3 (t = 0 k 4 (t = 0 k 1 (t = - t 2 k 2 (t = 0.5t k 3 (t = t k 4 (t = - 0.5t 2 Time-varying Lorenz system dx/dt = -10(x - y + k 1 (t! y dy/dt = 28x - y - xz + k 2 (t! z dz/dt = xy - (8/3z + [k 3 (t + k 4 (t]! y

19 Uncovering full topology of oscillator networks (1 A class of commonly studied oscillator -network models: dx N i = F i (x i +! C ij (x j - x i (i = 1,..., N dt j=1,j!i - dynamical equation of node i N - size of network, x i " R m, C ij is the local coupling matrix C ij = C ij 1,1 C ij 2,1 C ij 1,2 C ij 2,2! C ij 1,m! C ij 2,m " " " " C ij m,1 C ij m,2! C ij m,m ( ( ( ( ( ' - determines full topology If there is at least one nonzero element in C ij, nodes i and j are coupled. Goal: to determine all F i (x i and C ij from time series.

20 X =! " x 1 x 2... x N Uncovering full topology of oscillator networks (2 Nm'1 ( Network equation is dx dt [G(X] i = F i (x i + = G(X, where (x j - x i N C ij j=1,j!i A very high-dimensional (Nm-dimensional dynamical system; For complex networks (e.g, random, small-world, scale-free, node-to-node connections are typically sparse; In power-series expansion of [G(X] i, most coefficients will be zero - guaranteeing sparsity condition for compressive sensing. W.-X. Wang, R. Yang, Y.-C. Lai, V. Kovanis, M. A. F. Harrison, Time-series based prediction of complex oscillator networks via compressive sensing, Europhysics Letters 94, (2011.

21 Evolutionary-game dynamics Example: Prisoner s dilemma game! Strategies: cooperation S(C = "! Payoff matrix: P(PD = " 1 0 b 0 1 0! ; defection S(D = " b - parameter Payoff of agent x from playing PDG with agent y: For example, M C'C = 1 M D'D = 0 M C'D = 0 M D'C = b 0 1 M x'y = S x T PS y

22 Evolutionary game on network (social and economical networks A network of agents playing games with one another:! Adjacency matrix = " a xy : Payoff of agent x from agent y: M x*y = a xy S x T PS y ' a xy =1 if x connects with y ( a xy = 0 if no connection Time series of agents (Detectable (1 payoffs (2 strategies Compressive sensing Full social network structure

23 Prediction as a CS Problem Payoff of x at time t: M x (t = a x1 S x T (tps 1 (t + a x2 S x T (tps 2 (t +!+ a xn S x T (tps N (t! Y = "! ' = " M x (t 1 M x (t 2! M x (t m! X = " a x1 a x2! a xn S x T (t 1 PS 1 (t 1 S x T (t 1 PS 2 (t 1! S x T (t 1 PS N (t 1 S x T (t 2 PS 1 (t 2 S x T (t 2 PS 2 (t 2! S x T (t 2 PS N (t 2!!!! S x T (t m PS 1 (t m S x T (t m PS 2 (t m! S x T (t m PS N (t m X : connection vector of agent x (to be predicted W.-X. Wang, Y.-C. Lai, C. Grebogi, and J.-P. Ye, Network reconstruction based on evolutionary-game data, submitted Y = ' X Y,': obtainable from time series Compressive sensing a a X = a x1 x2 xn x y N matching Full network structure Neighbors of x

24 Success rate for model networks

25 22 students play PDG together and write down their payoffs and strategies Friendship network Reverse engineering of a real social network Experimental record of two players Observation: Large-degree nodes are not necessarily winners

26 20 Detecting hidden nodes (a Idea Two green nodes: immediate neighbors of hidden node Information from green nodes is not complete Anomalies in the prediction of connections of green nodes Node Node (b Node Node Directed/weighted network: adjacency matrix not symmetric Node (c 2 Variance of predicted coefficients

27 Distinguishing between effects of hidden node and noise ( Say, h - hidden node; i, j - two neighboring nodes of h Variance of predicted coefficients Hidden node: as N m is increased through a threshold value, variance of predicted coefficients of i and j will decrease 16 7 Noise - variance will not change 2. Cancellation factor w ih! weight of link between i and h; similarly for w jh Cancellation factor:! ij = w ih / w jh Hidden node:! ij will change and then plateaued Noise:! ij will remain approximately constant

28 Distinguishing between effects of hidden node and noise (2 1 (a Hidden 10 5 (b 0.5 Noise Hidden ij Noise (c R m (d R m R.-Q. Su, W.-X. Wang, Y.-C. Lai, and C. Grebogi, Reverse engineering of complex networks: detecting hidden nodes, submitted

29 Limitations (1 1. Key requirement of compressive sensing the vector to be determined must be sparse. Dynamical systems - three cases: Vector field/map contains a few Fourier-series terms - Yes Vector field/map contains a few power-series terms - Yes Vector field /map contains many terms not known Ikeda Map: F(x, y= [A + B(x cos!! ysin!, B(xsin! + ycos!] where! " p! k - describes dynamics in an optical cavity 1+ x y Mathematical question: given an arbitrary function, can one find a suitable base of expansion so that the function can be represented by a limited number of terms?

30 Limitations (2 2. Networked systems described by evolutionary games Yes 3. Measurements of ALL dynamical variables are needed. Outstanding issue If this is not the case, say, if only one dynamical variable can be measured, the CS-based method would not work. Delay-coordinate embedding method? - gives only a topological equivalent of the underlying dynamical system (e.g., Takens embedding theorem guarantees only a one-to-one correspondence between the true system and the reconstructed system. 4. In Conclusion, Much work is needed!

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