KTH/CSC March 10, 2010Erik Aurell, KTH & Aalto University1 Dynamic cavity method and marginals of non-equilibrium stationary states KITPC/ITP-CAS Program

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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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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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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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..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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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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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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…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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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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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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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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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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The cavity method and BP output for spin histories

Cavity graph

The BP assumption: spins in cavity independent

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BP updates for histories

The complexity is O(N×L×). Hence hard to use unless furtherapproximations are made.

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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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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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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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Comparing dynamic cavity equations to Monte Carlo

…for more, Hamed Mahmoudi’s talk in next

week’s workshop…

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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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Half-asymmetric networks

Sequential

Connectivity c=3

Size N=1000


Local fields ≈ 0.01

Couplings ≈ 1⁄√𝑐

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Symmetric networks

Sequential

Connectivity c=3

Size N=1000


Local fields ≈ 0.01Couplings ≈

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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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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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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

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Thanks to Hamed Mahmoudi

John Herz

Yasser Roudi

Hong-Li Zeng

Mikko Alava

Izaac Neri

Lenka Zdeborová

Silvio Franz

Documents

KTH/CSC March 10, 2010Erik Aurell, KTH & Aalto University1 Dynamic cavity method and marginals of non-equilibrium stationary states KITPC/ITP-CAS Program