physcom-networks: what does physics say about the design

Supelec

Alcatel-Lucent Chair on Flexible Radio

Physcom-networks: What does physics sayabout the design of wireless networks?

Merouane Debbahmerouane.debbah@supelec.fr

The role of Physcom-networks

• The aim is to describe dense networks in terms of macroscopic parameters rather thanin terms of microscopic parameters using tools from Physics.

• These macroscopic quantities retain just enough information to allow meaningful andtractable problems for the routing optimization of the network.

• We will provide two examples:We solve the deployment design problem of sensor networksWe solve the routing problem in networks for affine costs per packet.

1

First Part, joint work with Ø. Ryan

Deployment design

Ø. Ryan and M. Debbah, ”Random Vandermonde Matrices-Part I: Fundamental results”,submitted to IEEE transactions on Information theory, 2008

Ø. Ryan and M. Debbah, ”Random Vandermonde Matrices-Part II: Applications”,submitted to IEEE transactions on Information theory, 2008

2

Signal reconstruction

We would like to study the optimal distribution of sensor networks.

The nodes of a sensor network can be used to sample a physical field like

• air temperature• light intensity• pollution levels...

and report data to a common processing unit (sink node).

We evaluate the impact of the distribution of the sensors on the quality of thereconstructed signal.

3

Signal reconstruction

Consider a unidimensional bandlimited physical field, described by its harmonics (s is thespace index):

y(s) =1√N

N−1∑

k=0

xke−j2πks

N

The field is sampled in the space domain by L sensors which are deployed in thenormalized interval [0, 1] in positions [s1, ..., sL] with si ∈ [0, 1] (ωi =

2πsiN ).

y(ωi) =1√N

N−1∑

k=0

xke−jkωi.

The task of the reconstruction algorithm is to calculate an estimate x of the spectrum x inthe presence of noise.

The distribution of the deployment of the sensors affects the performance of signalreconstruction in the presence of noise

4

Signal reconstruction

With noise:

y = VTx + n

with y = [y(ω1), ...y(ωL)]T , x = [x1, ..., xN ]T ,n = [n1, ..., nL] and

V =1√N

1 · · · 1

e−jω1 · · · e−jωL

... . . . ...e−j(N−1)ω1 · · · e−j(N−1)ωL

. (1)

MMSE =1

NE || x− x ||2→

∫g(λ)dF (λ)(Guo, Verdu, Shamai)

The MMSE is closely related to the empirical eigenvalue distribution of VVH i.edF (λ) = 1

N

∑Ni=1 δ(λi − λ) where λi are the eigenvalues of VVH.

5

Our Framework

We will consider Random Vandermonde matrices V of dimension N × L of the form

V =1√N

1 · · · 1

e−jω1 · · · e−jωL

... . . . ...e−j(N−1)ω1 · · · e−j(N−1)ωL

(2)

where ω1,...,ωL are independent and identically distributed (phases) taking values on[0, 2π).

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”Memoire sur l’elimination”, Alexandre Vandermonde(1772)

V =

1 · · · 1

α1 · · · αL... . . . ...αN−1

1 · · · αN−1L

The determinant of a square Vandermonde matrix (L = N ) can be expressed as:

det(V) =∏

1≤i<j<n

(αj − αi)

Quite remarkably, this is the only thing we know about Vandermonde matrices.

7

A physics point of view: Schrondinger’s equation

HΦi = EiΦi

Φi is the wave functionEi is the energy levelH is the hamiltonian

Magnetic interactions between the spins of electrons

8

Asymptotic Random Matrix Theory

Eugene Paul Wigner, 1902-1995

9

Randomness in 1955

E. Wigner. ”Characteristic Vectors of bordered matrices with infinite dimensions”, Theannal of mathematics, vol. 62, pp.546-564, 1955.

1√n

0 +1 +1 +1 −1 −1

+1 0 −1 +1 +1 +1

+1 −1 0 +1 +1 +1

+1 +1 +1 0 +1 +1

−1 +1 +1 +1 0 −1

−1 +1 +1 +1 −1 0

As the matrix dimension increases, what can we say about the eigenvalues (energylevels)?

10

Wigner Matrices: the semi-circle law

11

The empirical eigenvalue distribution of H

H is Hermitian

dFN(λ) =1

N

N∑

i=1

δ (λ− λ1)

The moments of this distribution are given by:

mN1 =

1

Ntr (H) =

1

N

N∑

i=1

λi =

∫λdFN(λ)

mN2 =

1

Ntr (H)

2=

∫λ

2dFN(λ)

... = ...

mNk =

1

Ntr (H)

k=

∫λ

kdFN(λ)

In many cases, all the moments converge. This is exactly the type of results needed tounderstand the network.

12

Wigner Matrices: the semi-circle law

Wigner’s proof of the convergence to the semi-circle law:

The empirical moment 1NTrace(H2k) → The Catalan numbers

limN→∞

1

NTrace(H2k

) =

∫ 2

−2

x2k

f(x)dx

=1

k + 1C

2kk

Since the semi-circle law is symmetric, the odd moments vanish.

13

Wigner Matrices: the semi-circle law

Calculus based on recursion:

We integrate by parts and get:

α2k =1

π

∫ 2

−2

x2k

√4− x2dx

= − 1

2π

∫ 2

−2

−x√4− x2

x2k−1

(4− x2)dx

=1

2π

∫ 2

−2

√4− x2(x

2k−1(4− x

2))′dx

= 4(2k − 1)α2k−2 − (2k + 1)α2k

In this way, the recursion is obtained:

α2k =2(2k − 1)

k + 1α2k−2

14

Catalan Numbers

Eugene Charles Catalan, 1814-1894

15

Wigner Matrices: the semi-circle law

E. Wigner. ”On the Distribution of Roots of certain symmetric matrices”, The Annals ofMathematics, vol. 67, pp.325-327, 1958.

Theorem2. Consider a N ×N standard Wigner matrix W such that, for some constant κ

and sufficiently large N ,maxi,jE(| wij |4) ≤

κ

N2

Then the empirical distribution of W converges almost surely to the semi-circle law whosedensity is:

f(x) =1

2π

√4− x2

with | x |≤ 2

The semi-circle law is also known as the non-commutative analog of the Gaussiandistribution.

16

The empirical eigenvalue distribution of HHH

H is N ×K i.i.d Gaussian with KN = α

dFN(λ) =1

N

N∑

i=1

δ (λ− λ1)

The moments of this distribution are given by:

mN1 =

1

Ntr

(HHH

)=

1

N

N∑

i=1

λi → 1

mN2 =

1

Ntr

(HHH

)2

=1

N

N∑

i=1

λ2i → 1 + α

mN3 =

1

Ntr

(HHH

)3

=1

N

N∑

i=1

λ3i → α

2+ 3α + 1

17

The Marchenko-Pastur Distribution Law

V. A. Marchenko and L. A. Pastur, ”Distributions of eigenvalues for some sets of randommatrices,” Math USSR-Sbornik, vol.1 pp.457-483, 1967.

Theorem. Consider an N ×K matrix H whose entries are independent zero-meancomplex (or real) random variables with variance 1

N and fourth moments of order O( 1N2).

As K, N →∞ with KN → α, the empirical distribution of HHH converges almost surely

to a nonrandom limiting distribution with density

f(x) = (1− 1

α)+δ(x) +

√(x− a)+(b− x)+

2παx

where a = (1−√α)2 and b = (1 +√

α)2.

18

How do we deal with our signal reconstruction problem?

We denote by P(n) the set of all partitions of {1, ..., n}, and we will use ρ as notation fora partition in P(n). The set of partitions will be equipped with the refinement order ≤, i.e.ρ1 ≤ ρ2 if and only if any block of ρ1 is contained within a block of ρ2. Also, we will writeρ = {ρ1, ..., ρk}, where ρj are the blocks of ρ, and let |ρ| denote the number of blocks inρ

Definition 1. Define

Kρ,ω,N = 1

Nn+1−|ρ|×∫

(0,2π)|ρ|∏n

k=11−e

jN(ωb(k−1)−ωb(k))

1−ej(ωb(k−1)−ωb(k))

,

dω1 · · · dω|ρ|,

(3)

where ωρ1, ..., ωρ|ρ| are i.i.d. (indexed by the blocks of ρ), all with the same distribution

as ω, and where b(k) is the block of ρ which contains k (where notation is cyclic, i.e.b(−1) = b(n)). If the limit

Kρ,ω = limN→∞

Kρ,ω,N

exists, then Kρ,ω is called a Vandermonde mixed moment expansion coefficient.

19

Result

mn = limN→∞

E[trL

((VHV

)n)]

m1 = K1

m2 = K2 + K1,1c

m3 = K3 + K2,1c2 + K1,1,1c

2

m4 = K4 + K3,1c + K2,2c+

K2,1,1c2 + K1,1,1,1c

3

m5 = K5 + K4,1c + +K3,2c

+ K3,1,1c2 + K2,2,1c

2

+ K2,1,1,1c3 + K1,1,1,1,1c

4

One can show that the uniform case is related to the non-uniform case by:

Kρ,ω = Kρ,u(2π)|ρ|−1

(∫ 2π

0

pω(x)|ρ|

dx

).

20

Simulations: Gaussian versus uniform for β = 1c = N

L

21

Second Part, joint work with E. Altman and A. Silva

Routing design

E. Altman, M. Debbah and Alonso Silva, ”Continuum Equilibria for Routing in DenseAd-hoc Networks” 45th Annual Allerton Conference on Communication, Control andComputing, 2007.

22

Network Model

Assumption: The network is massively dense.

Let us consider the plane X1 ×X2 (on which are distributed sources and sinks ofinformation):The node density function d(x1, x2) [nodes/m2], such that the total number of nodes on aregion A, is then given by

N(A) =

∫

A

d(x1, x2)dS.

23

Network Model

The information density function ρ(x1, x2) [bps/m2]:

• If ρ(x1, x2) > 0 then there is a distributed data source.

• If ρ(x1, x2) < 0 then there is a distributed data sink.

We assume that the data created is equal to the data absorbed, i.e.∫

X1×X2

ρ(x1, x2)dS = 0.

24

Network Model

Figure 1: The function T.

The traffic flow function T(x1, x2) = (T1(x1, x2), T2(x1, x2))[bps/m], such that:

• Its direction coincides with the direction of the flow of information at point (x1, x2).• |T(x1, x2)| is the rate with which information crosses a linear segment perpendicular

to T(x1, x2) centered on (x1, x2).

25

The conservation equation

Over a surface Φ0 ⊆ X1×X2 of arbitrary shape, we assume that the information createdis equal to the information leaving the area, i.e.

∫

Φ0

ρ(x1, x2)dS =

∮

∂Φ0

[T · n(s)]dS

where the vector n(s) is the unit normal vector perpendicular to ∂Φ0 at a boundary point∂Φ0(s) and pointing outwards.

From the conservation equation holding for any smooth domain, then

∇ · T(x) :=∂T1(x)

∂x1

+∂T2(x)

∂x2

= ρ(x).

How to solve this problem and find the optimal route (flow of information) for a given nodedenstiy?

26

A physics point of view: Maxwell equations

div(E) =ρ

ε

div(B) = 0

rot(E) = −dBdt

rot(B) = µJ + µεdEdt

27

Gauss’s law

div(E) =ρ

ε

The optimal traffic flow is the same as the electrostatic field that will be created if weremove the data sources and substitute them with positive charges and we remove thedata sinks and substitute them with negative charges.

The cost functions will define the boundary conditions.

For some cost functions (minimize the number of nodes), one can show even that thetraffic is irrotational (Toumpis)

rot(E) = 0

28

Cost function

Considering g(x, T(x)) as a local cost function at point x, then the optimization problemis reduced to:

minimize Z over the flow distributions T

Z =

∫

Φ

g(x, T(x))dx1dx2 subject to

∇ · T(x) = ρ(x), ∀x ∈ Φ.

Kuhn-Tucker conditions implies for i = 1, 2:

∂g(x, T)

∂Ti

+∂ζ(x)

∂xi

= 0 if Ti(x) > 0

∂g(x, T)

∂Ti

+∂ζj(x)

∂xi

≥ 0 if Ti(x) = 0.

where the ζ(x) are Lagrange multipliers.

29

Cost function

It follows also that necessarily

ζ(x) = 0 ∀x ∈ ∂Φ where T(x) > 0.

This will provide in some cases the boundary condition to recover ζ.

Example : Affine cost per packet

Let the local cost function be

g(x, T(x)) =∑

i=1,2

gi(x, T(x))Ti(x).

gi(x, T(x)) =1

2ki(x)Ti(x) + hi(x).

30

Cost function

Then the Kuhn-Tucker conditions simplify to

ki(x)Ti(x) + hi(x) +∂ζ(x)

∂xi

= 0 if Ti(x) > 0

ki(x)Ti(x) + hi(x) +∂ζ(x)

∂xi

≥ 0 if Ti(x) = 0.

Assume ki(·) > 0. Let ai := 1/ki, and b s.t. bi := hi/ki. Assume that there exists asolution where T (x) > 0 for all x. Then

Ti(x) = −(

ai(x)∂ζ(x)

∂xi

+ bi(x)

).

31

Cost function

The function ζ(·) can be found as the solution in H10(Φ) of the elliptic equation

∑

i

∂

∂xi

(ai(x)

∂ζ

∂xi

)+∇·b(x) + ρ(x) = 0 .

Well behaved Dirichlet problem, known to have a unique solution in H10(Φ) (J. L. Lions).

32

Cost function

0 50 1000

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0 50 1000

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0 50 1000

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Figure 2: Minimum cost routes (cost = distance2) where relaynodes are placed according to a spatial Poisson process of densityλ(x, y) = a · (10−4x2 + 0.05) nodes/m2, for four increasing values of a.

33

Conclusion

A number of research groups are working independently on the connections betweenphysics and networks in the realm of massively dense nodes (optico-networks,magnetons,...).

The physics approach is a useful tool to get a hint on the macroscopic behavior.

34

Last Slide

THANK YOU!

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