Problem 1: Can Catastrophic Events in Dynamical Systems be ...
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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?
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Problem 2: Reverse Engineering of Complex Networks
NetworkMeasuredTime Series
Full network topology?Assumption: all nodes are
externally accessible
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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? How can we distinguish hidden node from local noise sources?
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Basic Idea (1)Dynamical system: dx/dt = F(x), x ∈ Rm Goal: to determine F(x) from measured time series x(t)!Power-series expansion of jth component of vector field F(x)
[F(x)] j = ... (a jlm=0
n
∑l2=0
n
∑l1=0
n
∑ )l1l2 ...lmx1l1x2
l2 ....xmlm
xk − kth component of x; Highest-order power: n(a j)l1l2 ...lm
- coefficients to be estimated from time series
- (1+n)m coefficients altogetherIf F(x) contains only a few power-series terms, most of the coefficients will be zero.
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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 = (1+3)3 = 64 unknown coefficients[F(x)]1 = (a1)0,0,0x
0y0z0 + (a1)1,0,0x1y0z0 + ... + (a1)3,3,3x
3y3z3
Coefficient vector a1=
(a1)0,0,0
(a1)1,0,0
...(a1)3,3,3
!
"
#####
$
%
&&&&&
- 64×1
Measurement vector g(t) = [x(t)0y(t)0z(t)0, x(t)1y(t)0z(t)0, ... , x(t)3y(t)3z(t)3] 1 × 64So [F(x(t))]1 = g(t)•a1
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Basic Idea (3)Suppose x(t) is available at times t0,t1,t2,...,t10 (11 vector data points)dxdt
(t1) = [F(x(t1))]1 = g(t1)•a1
dxdt
(t2 ) = [F(x(t2 ))]1 = g(t2 )•a1
...dxdt
(t10 ) = [F(x(t10 ))]1 = g(t10 )•a1
Derivative vector dX =
(dx/dt)(t1)(dx/dt)(t2 )
...(dx/dt)(t10 )
!
"
#####
$
%
&&&&&
10×1
; Measurement matrix G =
g(t1)g(t2 )
g(t10 )
!
"
#####
$
%
&&&&&
10×64
We finally have dX = G•a1 or dX10×1 = G10×64 • (a1)64×1
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Basic Idea (4) dX = G•a1 or dX10×1 = G10×64 • (a1)64×1
Reminder: a1 is the coefficient vector for the first dynamical variable x.To obtain [F(x)]2, we expand[F(x)]2 = (a2 )0,0,0x
0y0z0 + (a2 )1,0,0x1y0z0 + ... + (a2 )3,3,3x
3y3z3
with a2, the coefficient vector for the second dynamical variable y. We havedY = G•a2 or dY10×1 = G10×64 • (a2 )64×1
where
dY =
(dy/dt)(t1)(dy/dt)(t2 )
...(dy/dt)(t10 )
"
#
$$$$$
%
&
'''''
10×1
.
Note: measurement matrix G is the same.Similar expressions can be obtained for all components of the velocity field.
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Compressive Sensing (1)
Look at dX = G•a1 or dX10×1 = G10×64 • (a1)64×1
Note that a1 is sparse - Compressive sensing!
Data/Image compression:Φ : 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 = Φ•x
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Compressive Sensing (2)
Why l1 − norm? - Simple example in three dimensions
Find a vector x with minimum number of entries subject to the constraint y = Φ•x: l1 − norm
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
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Predicting Catastrophe (1)
Henon map: (xn+1, yn+1) = (1- axn2 + yn, bxn )
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 imminentbased on a limited set of measurements?
Step 1: Predicting system equations
Distribution of predicted values of ten power-series coefficients:constant, y, y2, y3, x, xy, xy2, x2, x2y, x3 # of data points used: 8
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Predicting Catastrophe (2)
Step 2: Performing numerical bifurcation analysis
BoundaryCrisisCurrent operation point
W.-X. Wang, R. Yang, Y.-C. Lai, V. Kovanis, and C. Grebogi, Physical Review Letters 106, 154101 (2011).
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Reconstructing Full Topology of Oscillator Networks (1)
A class of commonly studied oscillator -network models:dxidt
= Fi (x i ) + Cijj=1,j≠i
N∑ • (x j - x i ) (i = 1, ... , N)
- dynamical equation of node i N - size of network, xi∈ Rm, Cij is the local coupling matrix
Cij =
Cij1,1 Cij
1,2 Cij1,m
Cij2,1 Cij
2,2 Cij2,m
Cij
m,1 Cijm,2 Cij
m,m
#
$
%%%%%
&
'
(((((
- determines full topology
If there is at least one nonzero element in Cij, nodes i and j are coupled.
Goal: to determine all Fi(xi) and Cij from time series.
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Reconstructing Full Topology of Oscillator Networks (2)
X =
x1
x2
...xN
!
"
#####
$
%
&&&&&Nm×1
−Network equation is dXdt
= G(X), where
[G(X)]i = Fi (x i ) + Cijj=1,j≠i
N∑ • (x j - x 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”, EurophysicsLetters 94, 48006 (2011).
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Evolutionary-Game Dynamics
Strategies: cooperation S(C) = 10
!
"#
$
%&; defection S(D) = 0
1
!
"#
$
%&
Payoff matrix: P(PD) = 1 0b 0
!
"#
$
%& b - parameter
Payoff of agent x from playing PDG with agent y: M x←y = SxTPSy
For example, MC←C = 1 MD←D = 0 MC←D = 0 MD←C = b
Prisoner’s dilemma game
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Evolutionary Game on Network (Social and Economical Systems)
Full social network structureCompressive sensing
Time series ofagents (Detectable)
(1) payoffs(2) strategies
A network of agents playing games with one another:
Adjacency matrix =... ... ...... axy ...
... ... ...
!
"
###
$
%
&&&
: axy =1 if x connects with yaxy = 0 if no connection
'()
Payoff of agent x from agent y: M x←y = axySxTPSy
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Prediction as a CS Problem
2
1
Û
÷÷÷÷÷
ø
ö
ççççç
è
æ
=
xN
x
x
a
aa
!X
x
…
Neighbors of x
y…
+ ++!N
…
Full network structurematching
Compressive sensing
Payoff of x at time t: Mx (t) = ax1SxT (t)PS1(t)+ ax2Sx
T (t)PS2 (t)++ axNSxT (t)PSN (t)
Y =
Mx (t1)Mx (t2 )
Mx (tm )
!
"
#####
$
%
&&&&&
X =
ax1ax2
axN
!
"
#####
$
%
&&&&&
X : connection vector of agent x (to be predicted)
Φ =
SxT (t1)PS1(t1) Sx
T (t1)PS2 (t1) SxT (t1)PSN (t1)
SxT (t2 )PS1(t2 ) Sx
T (t2 )PS2 (t2 ) SxT (t2 )PSN (t2 )
SxT (tm )PS1(tm ) Sx
T (tm )PS2 (tm ) SxT (tm )PSN (tm )
!
"
#####
$
%
&&&&&
Y = Φ•X Y,Φ: from time series
• W.-X. Wang, Y.-C. Lai, C. Grebogi, and J.-P. Ye,“Network reconstruction based on evolutionarygame data,” Physical Review X 1, 021021, 1-7 (2011).
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Reverse Engineering of a Real Social Network
Friendship network
Experimental record of two players22 students play PDG together andwrite down their payoffs and strategies
Observation: Large-degree nodes are not necessarily winners
W.-X. Wang, Y.-C. Lai, C. Grebogi, and J.-P. Ye, “Network reconstruction based on evolutionary game data,”Physical Review X 1, 021021 (2011).
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Detecting Hidden Node 20
3 1512
5
2
1
16
1819
4
7
14
911
6
8
17
13
10
Node #
Nod
e #
5 10 15
5
10
15
Node #
Nod
e #
5 10 15
5
10
15 10−20 1000
2
4
6
8
10
12
14
16
18
20
σ2
Nod
e #
−0.2
−0.1
0
0.1
0.2
0.3
0.4
−0.2
−0.1
0
0.1
0.2
0.3
0.4(c)
(a)
(b)Idea
• Two green nodes: immediateneighbors of hidden node
• Information from green nodesis not complete
• Anomalies in the prediction of connections of green nodes
• R.-Q. Su, W.-X. Wang, and Y.-C. Lai, “Detecting hidden nodes in Complex networks from time series,” Physical Review E 106, 058701R (2012).
• R.-Q. Su, Y.-C. Lai, X. Wang, and Y.-H. Do, “Uncovering hidden nodes inComplex networks in the presence of noise,” Scientific Reports 4, Article number 3944 (2014).
Variance of predicted coefficients
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Locating Hidden Source in Complex Spreading Network
Using Binary Time Series
• Z.-S. Shen, W.-X. Wang, Y. Fan, Z.-R. Di, and Y.-C. Lai, “Reconstructing propagation networks with natural diversity and identifying hidden source,” Nature Communications 5, Article number 4323 (2014).
• Z.-L. Hu, X. Han, Y.-C. Lai, and W.-X. Wang, “Optimal localization of diffusion sources in complex networks,” Royal Society Open Science 4, Article number 170091 (2017).
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(E)
Node #
Node
#
10 20 30
5
10
15
20
25
30
0
0.1
0.2
0.3
0.4
(C) Reconstructed Weight
Node #
Node
#
10 20 30
5
10
15
20
25
30
0
0.005
0.01
0.015
0.02
0.025
(D) Reconstructed Delays
... ...
X
X
X
1
2
N
(A)
t1 t2 tw
Time Series collectionFinal geo-spatial network (2D)
1
23
4
5
6
7
8
9
10
11
12
13
14
15 16
17 18
19
20
21
22
23 24
2526
27
28
29
30
Úxi(t) =xi (t + ¨ t) í xi(t í ¨ t)
2× ¨ t
(B) Compressive Sensing
56
2329
50
20
24
38
419
34
1611
32
37
143
33
9
47
10
22
28
42
15
26
14
12
49
7
44
39
8
21
3
30
27
13
35
25
40
2
41
1731
48
45
1846
36
(F) Final geo-spatial network (3D)
Reconstruction of complex geospatial networks
R.-Q. Su, W.-X. Wang, X. Wang, and Y.-C. Lai, “Data-based reconstruction of complex geospatial networks, nodal positioning and detection of hidden nodes,” Royal Society Open Science 3, 150577 (2016).
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Discussion
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
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?
Ikeda Map: F(x, y)= [A + B(xcosφ − ysinφ),B(xsinφ + ycosφ)]
where φ ≡ p− k1+ x2 + y2 - describes dynamics in an optical cavity
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Data Analysis and Complex Systems
Data-based Research is becoming increasingly important for understanding, predicting, and controlling Complex Dynamical Systems
Complex Dynamical Systems
Big or Small Data Analysis
Big Data or Small Data?
2. Networked systems described by evolutionary games – Yes3. Measurements of ALL dynamical variables are needed.
Outstanding issueIf 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).
Discussion