CS774. Markov Random Field : Theory and Application

Lecture 21

Kyomin JungKAIST

Nov 19 2009

Application of MRF in Wireless Network:Routing and Scheduling

In a wireless network, designing a simple and distributed routing/scheduling algorithm is of primary importance.

Algorithm determines where(routing) and when(scheduling) to send packets.

Due to wireless interference, the scheduling at each time must satisfy certain constraints.

MaxWeight algorithm based on MWIS and its variants are widely used.

Remind: MRF forMaximum Weight Independent Set (MWIS)

Given a graph G=(V,E), a subset I of V is called an Indepen-dent Set, if for all , the two end points of e does not belong to I simultaneously.

When the vertices are weighted, an independent set I is called MWIS if the sum of the weights of is maximum.

Finding a MWIS is equivalent to finding a MAP in the following MRF on

where , and

Evuvu

Vvvv xxxWxXP

),(

),,(exp][

vW

||}1,0{ VX

),( 21 xx 0

1

121 xxif

otherwise

is the weight at node v.

Ee

Iv

The Network Model

Time is slotted.

At each time, per-destination packets arrive to the network. unicast, multicast

Queues for separate destinations.

Algorithm decides routing and scheduling.

Interference Constraint

We say a pair of links forms an interference if the two links cannot transmit packets at the same time.

Interference set is the set of all those pairs.

Interference Constraint

We say a pair of links forms an interference if the two links cannot transmit packets at the same time.

Interference Constraint

This model includes any generally considered interference model Ex. K-hop, node exclusive… Can be generalized to directed

network graph

Consider a graph G’=(L, I) where L is the links, and I is the interference set.

A subset L’ of L is feasible scheduling if and only if L’ is an independent set of G’.

MaxWeight Algorithm

5

1

4

100

2

Weight of a link is the difference of the queue sizes at the end nodes.

3

MaxWeight Algorithm

Computing MaxWeight is equivalent to computing Max Weight Independent Set (MWIS) in G’=(L,I).

5

1

4

100

2

3

MaxWeight Algorithm

MaxWeight is very well stud-ied. L. Tassiulas, A. Ephremides, IEEE

Transactions on Information the-ory 1992.

B. Awerbuch, T. Leighton, STOC 1994.

+ 100s of other papers

Provides adaptive routing and scheduling.

For some cases of interference constraint, like edge con-straint, it is fully distributed.

Dynamics of the WaxWeight

The algorithm can be understood so that the following potential function decreases as much as possible.

Where is the height of the packet, and is the queue size.

L. Tassiulas, A. Ephremides, [’92] showed that the MaxWeight makes the system stable (queue size bounded over time) for iid sto-chastic packet arrivals with feasible arrival rate for any interference.

)||()( 2

QueuesqPacketsp

qphP

)( ph

h = 3

h = 1

|| q

MRF in Statistical Physics: The Ising Model

Consider a sheet of metal:

It has the property that at low temperatures it is magnetized, but as the temperature increases, the magnetism “melts away”. We would like to model this behavior. We make some simplifying assumptions to do so.

The individual atoms have a “spin”, i.e., they act like little bar magnets, and can either point up (a spin of +1), or down (a spin of –1).

Neighboring atoms with different spins have an interaction energy.

The atoms are arranged in a lattice.

Possible states of the Lattice

A choice of ‘spin’ at each lattice point.

2q

Ising Model has a choice of two

possible spins at each point

The Kronecker delta function and the Hamiltonian of a state

,

0 for

1 for a b

a b

a b

Kronecker delta-function is defined as:

The Hamiltonian of a system is the sum of the energies on

edges with endpoints having the same spins.

,edges

H J a b

where and are the endpoints of the edge, and is the energy of the edge.a b J

(J>0)

The energy (Hamiltonian) of the state

of this system is ( )H w -11J

,( )i js s

edgesH w J

0

10

0

000 0

00

0

1

1

1 111

1

11

1

0

0

00

0 0

0 0 0

0 Endpoints have different spins, so δ is 0.

Endpoints have the same spins, so δ is 1.

Ising Model at Different Temperatures

Cold Temperature Hot Temperature

Probability of a State Occurring

( )

all states

H w

H w

w

e

e

1

kT

231.38 10

, where T is the temperature and k is the Boltzman constant joules/Kelvin.

]Pr[ wState

The denominator is the partition function of the system.

Effect of Temperature Consider two different states A and B, with H(A) <

H(B). The relative probability that the system is in the two states is:

At high temperatures (i.e., for kT much larger than the energy difference |D|), the system becomes equally likely to be in either of the states A or B.

At lower temperatures, the system is more likely to be in the lower energy state.

( ) ( )

all states all states

H A H B

H A H B

w w

P A e e

P Be e

( )

( ), where 0.

DDH AkT kT

H B

ee e D H A H B

e

The Potts Model (generalized Ising Model)Now let there be q possible states.

2q 3q 4q

Orthogonal vectors

Ex) Colorings of the points with q colors

Applications of the Potts Model

● Liquid-gas transitions● Computer Vision● Protein Folding● Biological Membranes● Social Behavior● Separation in binary alloys● Spin glasses● Neural Networks● Community detection, etc

Potts Model is an widely used form of MRF model.