Federico RicciTersenghi, INFM Rome

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EVERGROW SP4

• Federico Ricci-Tersenghi, INFM Rome
• Partners INFM, Aston, ENS, ICTP, ISI, Orsay

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Inference and Optimizationon (sparse) Networks
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SP4 WorkPackages

• WP4a Beliefs and Surveys
• Inference and Optimization
• Constraint Satisfaction Problems SAT, coloring,
  ...
• Phase Transitions in Random Ensemble
• Tree-like structures and loopy networks

• WP4b Error Correcting Codes
• Reliably communicating over a noisy channel
• Optimal encoding/decoding algorithms
• Noise thresholds (infinite codeword length)
• Corrections for finite lengths

• WP4b has been merged in WP4a
• large overlap
• lower activity (milestones reached)

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SP4 2006 Highlights

• WP4a Beliefs and Surveys
• Reconstructing a broadcasted message the spin
  glass transition
• Diverging time/length scales in glassy systems
• Detailed description of the geometry of the space
  of solutions in CSP
• Finding long cycles in graphs
• More CSP solved (COL with max local diversity,
  1-in-3 SAT,...)
• ...and many new questions...
• WP4b Error Correcting Codes
• Phase transitions in LDPC codes
• Finite length optimization
• Error exponents for LDPC codes
• Optimizing detection in multiuser communications

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...and not only sparse random graphs

• Bethe approximation and message-passing
  algorithms
• are exact on trees
• typically work for very sparse graphs
• if improved, work for sparse loopy graphs (SP vs
  BP)
• and even on some dense graph, e.g.
• binary perceptron, by reinforcement
• CDMA, by replicated systems

• Real-world (non random) networks
• internet
• biological networks

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Reconstruction on trees (i)
Mezard, Montanari, J. Stat. Phys. 124 (2006) 1317
On a k-ary tree with noisy links, given the
configuration at the leaves, can we
reconstruct the signal broadcasted from the root?
Each link is a noisy channel
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Reconstruction on trees (ii)
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Reconstruction on trees (iii)
The boundary condition is random according
to the broadcast process
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...and the spin glass transition
Unconditional distributions satisfy 1RSB
equations with Parisi parameter m1
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A spin glass phase on a tree?

• Boundary conditions are correlated!
• Disregarding correlations in BC gives RS results
• Simple interpretation of RSB, allows rigorous
  derivations
• Point-to-Set correlation function

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Geometry of solutions-space

• Cavity resultsSee Lenkas talk
• Rigorous results (for random k-SAT)For
  and every solution belongs
  whp toa small cluster, where most of the
  variables are frozenNo proof for
  and
• Algorithmic implicationsFor Glauber dynamics we
  haveBut with message-passing (BP or SP) we can
  do better

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Finding (long) cycles in graphs

• In 2005 Marinari Semerjian computed the number
  of cyclesin a given graph by developing a
  specific MPA.
• The same algorithm decimation can find long
  cycles(short cycles are easy to find and
  enumerate).
• Finding Hamiltonian cycles is NP-complete.
• Comparison of 3 algorithms
• MPA, fast but less efficient for denser graphs
• MPA rewiring, fast and very efficient (gt 92)
• MCMC, a bit slower, many parameters, but perfect
  efficiency (100), at variance with random CSP.
• See Valerys talk for more details.

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Error exponents for LDPC codes
Prob(code,noise) with s exp-N L(s)
channel noise
0
pc
pd
s
For small noise most dangerous codes have
sub-exponential complexities (energetic large
deviation analysis)
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Plans for 2007

• Group testing by MPA (detection of failures based
  on local message exchanges in distributing
  computing)
• Dense-graph message passing algorithms for
  reverse engineering problems (inferring networks
  from measured data)
• Community detection on complex networks
  (Internet) by reinforced message passing
  applications
• Interpolating between easy and hard to solve
  problemsnew behavior close to the boundary?
• Effect of small scale structures (loops) on
  convergence of MPA
• Quantum ECC