#### Transcript Belief Propagation algorithm in Markov Random Fields

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] expWv xv ( xu , xv ), vV (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) pPackets ( | q | ) 2 qQueues 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. q2 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 q2 q3 q4 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.