Mezić Research Group
SIAM Conference on Applications of Dynamical SystemsMay 22-26, 2011 Snowbird, Utah
Ryan Mohr and Igor Mezić
LOW-DIMENSIONAL MODELS FOR
TCP-LIKE NETWORKS USING THE
KOOPMAN OPERATOR
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
Overview
2
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
Overview• Interested in characterizing network dynamics
Periodicity
Global coherent behavior etc.
2
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
Overview• Interested in characterizing network dynamics
Periodicity
Global coherent behavior etc.
• Build a low-dimensional model
2
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
Overview• Interested in characterizing network dynamics
Periodicity
Global coherent behavior etc.
• Build a low-dimensional model
• What is the appropriate way to compare model and data?
2
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
Overview• Interested in characterizing network dynamics
Periodicity
Global coherent behavior etc.
• Build a low-dimensional model
• What is the appropriate way to compare model and data?
2
Koopman Operator
Spectral Analysis
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
Overview• Interested in characterizing network dynamics
Periodicity
Global coherent behavior etc.
• Build a low-dimensional model
• What is the appropriate way to compare model and data?
2
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
Overview• Interested in characterizing network dynamics
Periodicity
Global coherent behavior etc.
• Build a low-dimensional model
• What is the appropriate way to compare model and data?
2
Koopman Delay Embeddings
Pseudo-metrics
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
Assumptions on the system• Discrete-time dynamical system
3
(X, T ) T : X → X
• Compact state space Variables for each router• buffer size
• congestion window
• roundtrip timer
• etc,
• Continuous Transformation TCP protocol
• On an attractor
• Moderately size network = large state space
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
Observables• Introduce observable on the system
assume observable is continuous, measurable
4
g : X → R
X
g
R
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
Observables• Introduce observable on the system
assume observable is continuous, measurable
4
g : X → R
• Choose functions that are relevant link loads for a subset of paths
load on a routers Traffic matrices
X
g
R
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
Koopman Operator, U• Interested in how the transformation drives the dynamics on
the real line
• Introduce the Koopman operator
Properties: Linear & Bounded (when integrating against a preserved measure)
• Interpretation1) Operator acting on space of functions = evolves observables, or2) For each point in phase space, get a time series in :
5
(Ukg)(x) := g(T kx)
Ryk = (Ukg)(x)
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
Spectral Decomposition of U• Want to decompose operator as action of projections onto
eigenspaces
• Let
6
U =�
j
λjPj
Uϕj(x) = λjϕj(x)
g ∈ span{ϕj} =⇒ g(x) =�
j
cjϕj(x)
=⇒ Ukg(x) =�
j
λkj cjϕj(x)
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
Spectral Decomposition of U• Want to decompose operator as action of projections onto
eigenspaces
• Let
• How do we compute the projection?
7
U =�
j
λjPj
Uϕj(x) = λjϕj(x)
g ∈ span{ϕj} =⇒ g(x) =�
j
cjϕj(x)
=⇒ Ukg(x) =�
j
λkj cjϕj(x)� �� �Pjg(x)
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
Harmonic Projections• Define class of operators
• The limit is an eigenfunction of the Koopman operator :
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Ug∗ω(x) = ei2πkωg∗ω(x)
Uk =�
ω
ei2πkωPωT
PωT g(x) := lim
n→∞
1n
n−1�
k=0
e−i2πkωUkT g(x) =: g∗ω(x)
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
Abilene Data Set*
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* http://www.cs.utexas.edu/~zhang/research/AbileneTM
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
Abilene Data Set*
• Data collected over 24 weeks
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* http://www.cs.utexas.edu/~zhang/research/AbileneTM
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
Abilene Data Set*
• Data collected over 24 weeks
• Sampling period of 5 minutes = 48,384 observations
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* http://www.cs.utexas.edu/~zhang/research/AbileneTM
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
Abilene Data Set*
• Data collected over 24 weeks
• Sampling period of 5 minutes = 48,384 observations
• Observable = Traffic Matrices
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* http://www.cs.utexas.edu/~zhang/research/AbileneTM
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
Abilene Data Set*
• Data collected over 24 weeks
• Sampling period of 5 minutes = 48,384 observations
• Observable = Traffic Matrices Measures the amount of traffic between input node and output node
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* http://www.cs.utexas.edu/~zhang/research/AbileneTM
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
Abilene Data Set*
• Data collected over 24 weeks
• Sampling period of 5 minutes = 48,384 observations
• Observable = Traffic Matrices Measures the amount of traffic between input node and output node
Quantities computed via some estimated procedure
9
* http://www.cs.utexas.edu/~zhang/research/AbileneTM
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
Abilene Data Set*
• Data collected over 24 weeks
• Sampling period of 5 minutes = 48,384 observations
• Observable = Traffic Matrices Measures the amount of traffic between input node and output node
Quantities computed via some estimated procedure Unfortunately, data for only 1 initial condition
9
* http://www.cs.utexas.edu/~zhang/research/AbileneTM
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
Abilene Data Set*
• Data collected over 24 weeks
• Sampling period of 5 minutes = 48,384 observations
• Observable = Traffic Matrices Measures the amount of traffic between input node and output node
Quantities computed via some estimated procedure Unfortunately, data for only 1 initial condition
Have 144 different observables
9
* http://www.cs.utexas.edu/~zhang/research/AbileneTM
Can change this on the Master Slide
Mezić Research Group
Monday, May 23, 2011 10
• Horizontal streaks = source/destination pair having continuous spectrum
• Vertical Streaks = majority of the network has a component at that frequency; globally relevant frequency
Can change this on the Master Slide
Mezić Research Group
Monday, May 23, 2011 11
gi(x)
Pωgi(x)
{ωj , Aj , θj}
yi(k) = A0 + 2D�
j=1
Aj cos(2πωjk + θj)
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
12
Building a Low-Dimensional Model
Harmonic Projections
Pω ∀ω ∈ [−0.5.0.5]
Cluster&
Order Modes
Extract first D modes
ω ∈ (0, 0.5)
Low-dimensional deterministic “skeleton”
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
Comparing Data and ModelTo Measure Closeness : Introduce family of pseudo-metrics to compare discretized measures in a delay space
13
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
Comparing Data and ModelTo Measure Closeness : Introduce family of pseudo-metrics to compare discretized measures in a delay space
13
Re
{yi(0), yi(1), . . . }πe
{
yi(0)
...yi(e− 1)
,
yi(1)
...yi(e)
, · · · }
Delay Embedding
• Delay Space:
≡ limk→∞
1n
n�
k=0
UkT (IBi ◦ πe ◦ g(x))
px,T (Bi) := P 0T (IBi ◦ πe ◦ g(x))
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
Comparing Data and Model• Computing measures in delay space :
Define a regular grid in delay space
Let be indicator function on a box in grid Empirical measure :
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≡ limk→∞
1n
n�
k=0
IBi(
y(k)
...y(k + e− 1)
)
IBi
≡ limk→∞
1n
n�
k=0
e−i2πωkUkT (IBi ◦ πe ◦ g(x))
≡ limk→∞
1n
n�
k=0
e−i2πωkIBi(
y(k)
...y(k + e− 1)
)
px,ω,T (Bi) := PωT (IBi ◦ πe ◦ g(x))
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
Comparing Data and Model• Computing measures in delay space :
Spectral measures :
15
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
Comparing Data and Model• Pseudo-metrics :
For each relevant frequency
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dx,ω(Td, Tm) =b�
i=1
|px,ω,Td(Bi)− px,ω,Tm(Bi)|
Dx(Td, Tm) = κ0dx,0(Td, Tm) +�
j
κjdx,ωj (Td, Tm)
{κj} = weights for each frequency
x(k) = 40 cos(2π
√2
20k + π/4) + 5 cos(2π
110
k)
e = 2 d0 = 0.5388
d0.0707 = 1.1766
D(Td, Tm) = 0.8573
Modes = 2
Can change this on the Master Slide
Mezić Research Group
Monday, May 23, 2011 17
50 40 30 20 10 0 10 20 30 40 5050
40
30
20
10
0
10
20
30
40
50Delay Space Trajectories
y(k)
y(k+
1)
DataModel
Toy Example
d0.0707 = 1.0933d0 = 0.0161
d0.1 = 0.443
D(Td, Tm) = 0.5175
Modes = 3
Can change this on the Master Slide
Mezić Research Group
Monday, May 23, 2011 18
50 40 30 20 10 0 10 20 30 40 5050
40
30
20
10
0
10
20
30
40
50Delay Space Trajectories
y(k)
y(k+
1)
DataModel
D(Td, Tm) = 0.8388
e = 2
Modes = 3
Can change this on the Master Slide
Mezić Research Group
Monday, May 23, 2011 19
Abilene
e = 2
Modes = 14
D(Td, Tm) = 0.2124
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Mezić Research Group
Monday, May 23, 2011 20
Abilene
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
21
A few comments
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
21
A few comments1. We presented the theory for deterministic dynamical systems;
everything carries through for the stochastic case.
(a) Define a random dynamical system and the stochastic Koopman operator.
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
21
A few comments1. We presented the theory for deterministic dynamical systems;
everything carries through for the stochastic case.
(a) Define a random dynamical system and the stochastic Koopman operator.
2. The analysis shows the network has deterministic period behavior, but also that the continuous portion of the spectrum is quite important to get the dynamics right.
Monday, May 23, 2011 Can change this on the Master Slide
Mezić Research Group
21
A few comments1. We presented the theory for deterministic dynamical systems;
everything carries through for the stochastic case.
(a) Define a random dynamical system and the stochastic Koopman operator.
2. The analysis shows the network has deterministic period behavior, but also that the continuous portion of the spectrum is quite important to get the dynamics right.
3. The variational norm for the measures needs to be replaced by a more suitable metric; the norm misbehaves when the discretization is refined and can get arbitrarily bad.