kth/csc march 10, 2010erik aurell, kth & aalto university1 dynamic cavity method and marginals...
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KTH/CSC
Erik Aurell, KTH & Aalto University 1March 10, 2010
Dynamic cavity method and marginals of non-equilibrium stationary states
KITPC/ITP-CAS ProgramInterdisciplinary Applications of Statistical Physics and
Complex Networks
E.A., H. Mahmoudi (2010) arXiv:1012.3388
I. Neri, D. Bollé, J. Stat Mech. (2009) P08009Y. Kanoria, A. Montanari (2009) arXiv:0907.0449
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Erik Aurell, KTH & Aalto University 2March 10, 2010
Motivation & outlineThe cavity method is a way to approximately compute marginals of equilibrium probability distributions – compare Belief Propagation (BP), iterative decoding.
Do these methods generalize from equilibrium statistical mechanics to non-equilibrium?
I will present results that this is (sometimes) possible.
Hamed Mahmoudi will tell more in the workshop next week.
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Erik Aurell, KTH & Aalto University 3March 10, 2010
BP and the Bethe-Peierls approximation
Marginal (one-spin) probabilities
Cost or energy function; sum of local terms
Configuration space; N discrete variables
JS Yedidia, WT Freeman, Y Weiss (2001)M Mézard, A Montanari, Oxford University Press (2009)
Belief propagationBethe-Peierls approximation
In the applications to statistical physics, artificial intelligence and information theory, such marginals (magnetizations, correlation functions etc) are (basically) what can be found by these methods
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Erik Aurell, KTH & Aalto University 4March 10, 2010
..messages, factor graphs...
• BP output equation
• BP update equation (one side)
• BP update equation (other side)
The “messages” and are Lagrange multipliers.
Exact if the factor graph is a tree. Otherwise no guarantees.
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Erik Aurell, KTH & Aalto University 5March 10, 2010
What then about non-equilbrium? Refined outline of the talk:
In non-equilibrium there is no Gibbs distribution. But is there nevertheless a message-passing scheme which (approximate) computes marginals of probability distributions? Compare dynamic mean-field theory.
• What could this be good for?• The example of kinetic Ising models• How dynamic cavity method works &
numerics
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Erik Aurell, KTH & Aalto University 6March 10, 2010
What could be the use?
Message-passing for non-equilibrium states should be much faster than Monte Carlo, but more accurate than mean field. However, on locally tree-like graphs.• Make bombs? Sorry, but lattice graphs
have lots of short loops – message-passing is not usually good.
• Games on graphs, perhaps states of social networks? Maybe, but that would not be entirely new.
Y. Kanoria, A. Montanari (2009) arXiv:0907.0449Majority dynamics on trees and the dynamic cavity method
…determining the stationary state of a Markov chain or a Markov process is a
rather general problem…
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Erik Aurell, KTH & Aalto University 7March 10, 2010
…the application we had in mind initially, but on which we so far have
done little…
(Potential) applications…
Another approach to kinetic inference by better computation of the marginals (magnetizations and correlations)
Roudi et al (2009)Hertz et al (2010)
Hertz, Roudi (PRL 2011)Zeng et al arXiv:1011.6216
Aree Witoelar in this program
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Erik Aurell, KTH & Aalto University 8March 10, 2010
Queuing theory looks for stationary states called “equilibrium states”.
Stationary states of networks of queues
Everitt (1994)Pallant, Taylor (1995)
Abdalla, Boucherie (2002)Vazquez-abad et al (2002)
Most of queuing theory is about networks of reversible or quasi-reversible queues, for which the stationary states are (relatively) simple. But there are also other examples e.g. modeling blocking probabilities in mobile cellular communication systems, with handovers between base stations.
source: Wikipedia
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Erik Aurell, KTH & Aalto University 9March 10, 2010
S Kauffman (1969)J Hopfield (1982)
B Derrida (1987)A Crisanti, H Sompolinski (1988)
….and many others Fully asynchronous updates (masterequations); not yet done
Synchronous updates; which can yet be varied in severalways (here in two ways)
- parallel : simultaneously update all spins- sequential : one spin updated at a time
x1 x2x3
X4xN
xi
The diluted asymmetric spin glass
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Erik Aurell, KTH & Aalto University 10March 10, 2010
Gaussian or binary
Starts as directed Erdős-Renyi graphs; where average connectivity would be cConnections i → j and j → i dependent
Coolen parametrization
Dilution, asymmetry, and interaction strength
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Erik Aurell, KTH & Aalto University 11March 10, 2010
MFT for magnetizations and correlations
Crisanti, Sompolinski (1988)Kappen, Spanjers (2000)
Hertz et al (2010)Hertz, Roudi arXiv:1009.5946
Zeng et al arXiv:1011.6216Hertz & Roudi (2011)
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Erik Aurell, KTH & Aalto University 12March 10, 2010
The cavity method and BP output for spin histories
Cavity graph
The BP assumption: spins in cavity independent
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Erik Aurell, KTH & Aalto University 13March 10, 2010
BP updates for histories
The complexity is O(N×L×). Hence hard to use unless furtherapproximations are made.
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Erik Aurell, KTH & Aalto University 14March 10, 2010
The time-factorized approximation
I Neri, D Bolle (2009)Aurell, Mahmoudi (2010)
exact for fully asymmetric
networks and parallel updates
…and for asymmetric networks the summation over spin i at time t-2 can be done.
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Erik Aurell, KTH & Aalto University 15March 10, 2010
The random sequential update model
When N goes to infinity, it is reasonable to expect that sequential update tends to fully asynchronous updates. But at finite N there will be a difference.
In random sequential updates (poor man’s Master equation) one (randomly picked) spin is updated at a time.
Consider first fully asynchronous updates (Master equations):
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Erik Aurell, KTH & Aalto University 16March 10, 2010
Dynamic cavity equations for sequential updates
Aurell, Mahmoudi (2010)
The full dynamic cavity equations:
The time factorized approximation (here not exact, even for fully asymmetric networks):
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Erik Aurell, KTH & Aalto University 17March 10, 2010
Comparing dynamic cavity equations to Monte Carlo
…for more, Hamed Mahmoudi’s talk in next
week’s workshop…
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Erik Aurell, KTH & Aalto University 18March 10, 2010
Aurell, Mahmoudi (2010)
Fully asymmetric networks
Sequential
Connectivity c=3
Size N=1000
MC averaged over 10 000 samples
Local fields ≈ 0.01
Couplings ≈
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Erik Aurell, KTH & Aalto University 19March 10, 2010
Aurell, Mahmoudi (2010)
Half-asymmetric networks
Sequential
Connectivity c=3
Size N=1000
MC averaged over 10 000 samples
Local fields ≈ 0.01
Couplings ≈ 1⁄√𝑐
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Erik Aurell, KTH & Aalto University 20March 10, 2010
Aurell, Mahmoudi (2010)
Symmetric networks
Sequential
Connectivity c=3
Size N=1000
MC averaged over 10 000 samples
Local fields ≈ 0.01Couplings ≈
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Erik Aurell, KTH & Aalto University 21March 10, 2010
Comparison to mean-field
Parallel updates
Connectivity c=3
Size N=10 000
MC averaged over 1000 samples
Local fields = 0.1
Ferromagnetic model,temperature =
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Erik Aurell, KTH & Aalto University 22March 10, 2010
Comparison to mean-field
Parallel updates“asymmetric ferromagnet”
Other parametersas previous slide
Dynamic BP and MC overlap also for finite time
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Erik Aurell, KTH & Aalto University 23March 10, 2010
Dynamic cavity method is computationally feasible within the time-factorized approximation. It is much faster than Monte Carlo.
It gives quite accurate results on the kinetic Ising models, both in the spin glass and in the ferromagnetic phases, and it is substantially more accurate than mean field.
It works (for now) for stationary states, not for transients.
Generalization to fully asynchronous dynamics lacking. Good applications are still to be developed.
Conclusions