R. Mohr: Low-dimensional Models for TCP-like Networks Using the Koopman Operator

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


Overview

2



Overview• Interested in characterizing network dynamics

Periodicity

Global coherent behavior etc.

2




Periodicity


• Build a low-dimensional model

2




Periodicity



• What is the appropriate way to compare model and data?

2




Periodicity




2

Koopman Operator

Spectral Analysis




Periodicity




2




Periodicity




2

Koopman Delay Embeddings

Pseudo-metrics



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



Observables• Introduce observable on the system

assume observable is continuous, measurable

4

g : X → R

X

g

R



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



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)



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)



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)



Harmonic Projections• Define class of operators

• The limit is an eigenfunction of the Koopman operator :

8

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)



Abilene Data Set*

9

* http://www.cs.utexas.edu/~zhang/research/AbileneTM

http://www.cs.utexas.edu/~zhang/research/AbileneTM




Abilene Data Set*

• Data collected over 24 weeks

9






Abilene Data Set*


• Sampling period of 5 minutes = 48,384 observations

9






Abilene Data Set*



• Observable = Traffic Matrices

9






Abilene Data Set*



• Observable = Traffic Matrices Measures the amount of traffic between input node and output node

9






Abilene Data Set*




Quantities computed via some estimated procedure

9






Abilene Data Set*




Quantities computed via some estimated procedure Unfortunately, data for only 1 initial condition

9






Abilene Data Set*




Quantities computed via some estimated procedure Unfortunately, data for only 1 initial condition

Have 144 different observables

9




Can change this on the Master Slide


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



Monday, May 23, 2011 11

gi(x)

Pωgi(x)

{ωj , Aj , θj}

yi(k) = A0 + 2D�

j=1

Aj cos(2πωjk + θj)



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”



Comparing Data and ModelTo Measure Closeness : Introduce family of pseudo-metrics to compare discretized measures in a delay space

13



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))



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 :

14

≡ 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))



Comparing Data and Model• Computing measures in delay space :

Spectral measures :

15



Comparing Data and Model• Pseudo-metrics :

For each relevant frequency

16

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



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



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



Monday, May 23, 2011 19

Abilene

e = 2

Modes = 14

D(Td, Tm) = 0.2124



Monday, May 23, 2011 20

Abilene



21

A few comments



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.



21




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.



21




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.

R. Mohr: Low-dimensional Models for TCP-like Networks Using the Koopman Operator

Documents