Belief Propagation algorithm in Markov Random Fields

Report
Nov 19 2009
CS774. Markov Random Field :
Theory and Application
Lecture 21
Kyomin Jung
KAIST
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 for
Maximum Weight Independent Set (MWIS)



Given a graph G=(V,E), a subset I of V is called an Independent Set, if for
all e  E , 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 v  I is maximum.
Finding a MWIS is equivalent to finding a MAP in the following
MRF on X {0,1}|V |


P[ X  x]  expWv  xv    ( xu , xv ),
 vV
 (u ,v )E
where ( x1 , x2 ) 

0
1
if x1  x2  1
otherwise
,
and
Wv is the weight at node v.
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 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
 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
Weight of a link is the difference of the queue sizes at the end nodes.
5
3
4
1
10
0
2
MaxWeight Algorithm
Computing MaxWeight is equivalent to computing Max
Weight Independent Set (MWIS) in G’=(L,I).
5
3
4
1
10
0
2
MaxWeight Algorithm
 MaxWeight is very well studied.
 L. Tassiulas, A. Ephremides, IEEE
Transactions on Information theory 1992.
 B. Awerbuch, T. Leighton, STOC 1994.
 + 100s of other papers
 Provides adaptive routing and
scheduling.
 For some cases of interference
constraint, like edge constraint, it is
fully distributed.
Dynamics of the WaxWeight

The algorithm can be understood so that the following
potential function decreases as much as possible.
P
 h( p)
pPackets
(
| q | )
2
qQueues
h=3
h=1
Where h( p) is the height of the packet, and | q | is the
queue size.

L. Tassiulas, A. Ephremides, [’92] showed that the
MaxWeight makes the system stable (queue size
bounded over time) for iid stochastic packet arrivals with
feasible arrival rate for any interference.
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.
q2
Ising Model has a
choice of two possible
spins at each point
The Kronecker delta function and the
Hamiltonian of a state
Kronecker delta-function is defined as:
0 for a  b
 a ,b  
1 for a  b
The Hamiltonian of a system is the sum of the energies on edges with
endpoints having the same spins.
H    J   a, b 
edges
where a and b are the endpoints of the edge, and J is the energy of the edge.
(J>0)
The energy (Hamiltonian) of the state
Endpoints have the same spins, so δ is 1.
1
0
0
1
1
0
0
0
1
0
1
0
0
0
0
0
1
1
1
0
1
0
Endpoints have different spins, so δ is 0.
0
1
0
0
0
0
0
0
1
H ( w)    J  si , s j
edges
H ( w) of this system is -11J
Ising Model at Different Temperatures
Cold Temperature
Hot Temperature
Probability of a State Occurring
Pr[State  w] 
e   H ( w)

e
  H  w
all states w
 
1
, where T is the temperature and
23
kT
k is the Boltzman constant 1.38  10 joules/Kelvin.
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:
P  A

P B
e   H ( A)

e
e  H ( B)
  H  A
all states w
  H ( A)
e
  H ( B)  e
e



e
 H  B
all states w

D
kT
D
 e kT
, where D  H  A  H  B   0.
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.
The Potts Model
(generalized Ising Model)
Now let there be q possible states.
Orthogonal vectors
q2
q3
q4
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.

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