Federico RicciTersenghi, INFM Rome

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Federico RicciTersenghi, INFM Rome

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Federico Ricci-Tersenghi, INFM Rome. Partners: INFM, Aston, ENS, ICTP, ... paramagnetic phase, no reconstruction. spin glass phase, reconstruction is. possible ... – PowerPoint PPT presentation

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Title: Federico RicciTersenghi, INFM Rome

1
EVERGROW SP4

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

2
Inference and Optimizationon (sparse) Networks
3
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)

4
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

5
...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

6
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
7
Reconstruction on trees (ii)
8
Reconstruction on trees (iii)
The boundary condition is random according
to the broadcast process
9
...and the spin glass transition
Unconditional distributions satisfy 1RSB
equations with Parisi parameter m1
10
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

11
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

12
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.

13
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)
14
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