1,342 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
Survey propagation at finite temperature: application to a Sourlas code as a toy model
In this paper we investigate a finite temperature generalization of survey
propagation, by applying it to the problem of finite temperature decoding of a
biased finite connectivity Sourlas code for temperatures lower than the
Nishimori temperature. We observe that the result is a shift of the location of
the dynamical critical channel noise to larger values than the corresponding
dynamical transition for belief propagation, as suggested recently by
Migliorini and Saad for LDPC codes. We show how the finite temperature 1-RSB SP
gives accurate results in the regime where competing approaches fail to
converge or fail to recover the retrieval state
The number of matchings in random graphs
We study matchings on sparse random graphs by means of the cavity method. We
first show how the method reproduces several known results about maximum and
perfect matchings in regular and Erdos-Renyi random graphs. Our main new result
is the computation of the entropy, i.e. the leading order of the logarithm of
the number of solutions, of matchings with a given size. We derive both an
algorithm to compute this entropy for an arbitrary graph with a girth that
diverges in the large size limit, and an analytic result for the entropy in
regular and Erdos-Renyi random graph ensembles.Comment: 17 pages, 6 figures, to be published in Journal of Statistical
Mechanic
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