Abstract
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 a wide class of random factor graph models. 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.
Highlights
Introduction and results1.1 Factor graphsIt is well known that viewing combinatorial optimization problems through the lens of Gibbs measures reveals important information about both structural and algorithmic aspects
Over the past few years there has been a great deal of interest in the Gibbs measures of random factor graph models
27:4 Belief Propagation on Replica Symmetric Random Factor Graph Models nodes x, y; μG,x,y is the distribution of the pair (σ(x), σ(y)) ∈ Ω2 for σ ∈ ΩV (G) chosen from the Gibbs measure
Summary
It is well known that viewing combinatorial optimization problems through the lens of Gibbs measures reveals important information about both structural and algorithmic aspects. We identify the set of all possible truth assignments with the Hamming cube {0, 1}n, and given a parameter β ≥ 0 we define functions ψi : {0, 1}n → (0, ∞) by letting ψβ,i(σ) = exp(−β 1{σ violates clause Φi}). By tuning β we can scan through the landscape on the Hamming cube defined by the function that maps each truth assignment to the number of clauses it leaves unsatisfied. This landscape has, a very substantial impact on the performance of algorithms. The bipartite graph gives rise to a metric on the set of variable and constraint nodes, namely the length of a shortest path.
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