5 research outputs found
Belief Propagation on replica symmetric random factor graph models
According to physics predictions, the free energy of random factor graph
models that satisfy a certain "static replica symmetry" condition can be
calculated via the Belief Propagation message passing scheme [Krzakala et al.,
PNAS 2007]. Here we prove this conjecture for two general classes of random
factor graph models, namely Poisson random factor graphs and random regular
factor graphs. Specifically, we show that the messages constructed just as in
the case of acyclic factor graphs asymptotically satisfy the Belief Propagation
equations and that the free energy density is given by the Bethe free energy
formula
An improved Belief Propagation algorithm finds many Bethe states in the random field Ising model on random graphs
We first present an empirical study of the Belief Propagation (BP) algorithm,
when run on the random field Ising model defined on random regular graphs in
the zero temperature limit. We introduce the notion of maximal solutions for
the BP equations and we use them to fix a fraction of spins in their ground
state configuration. At the phase transition point the fraction of
unconstrained spins percolates and their number diverges with the system size.
This in turn makes the associated optimization problem highly non trivial in
the critical region. Using the bounds on the BP messages provided by the
maximal solutions we design a new and very easy to implement BP scheme which is
able to output a large number of stable fixed points. On one side this new
algorithm is able to provide the minimum energy configuration with high
probability in a competitive time. On the other side we found that the number
of fixed points of the BP algorithm grows with the system size in the critical
region. This unexpected feature poses new relevant questions on the physics of
this class of models.Comment: 20 pages, 8 figure
The number of solutions for random regular NAE-SAT
Recent work has made substantial progress in understanding the transitions of
random constraint satisfaction problems. In particular, for several of these
models, the exact satisfiability threshold has been rigorously determined,
confirming predictions of statistical physics. Here we revisit one of these
models, random regular k-NAE-SAT: knowing the satisfiability threshold, it is
natural to study, in the satisfiable regime, the number of solutions in a
typical instance. We prove here that these solutions have a well-defined free
energy (limiting exponential growth rate), with explicit value matching the
one-step replica symmetry breaking prediction. The proof develops new
techniques for analyzing a certain "survey propagation model" associated to
this problem. We believe that these methods may be applicable in a wide class
of related problems
Belief Propagation on the random -SAT model
Corroborating a prediction from statistical physics, we prove that the Belief
Propagation message passing algorithm approximates the partition function of
the random -SAT model well for all clause/variable densities and all inverse
temperatures for which a modest absence of long-range correlations condition is
satisfied. This condition is known as "replica symmetry" in physics language.
From this result we deduce that a replica symmetry breaking phase transition
occurs in the random -SAT model at low temperature for clause/variable
densities below but close to the satisfiability threshold