Pascal talked about a connection of the Bethe entropy to the edge zeta function of a cycle code, building on previous work by Koetter, Li Vontobel and Walker. Starting with the first quantity, as is well known, Yedidia, Freeman and Weiss established that sum-product (if it converges) can only converge to fixed points of the Bethe free energy. The Bethe entropy is the non-linear term added in the ‘average energy’ (optimized by the LP decoder) and is therefore a fundamental quantity to understand sum-product and its connections to LP relaxations. A cycle code is a linear code where all the variables have degree two (and can be represented as edges).

Now what is the second quantity in this connection, the edge zeta function of a cycle code? As far as I understood (and this is probably inaccurate), for each edge (=variable of the code) define an indeterminate quantity .

For a primitive cycle  (without allowing backtracking) of this graph, consisting of edges , define a monomial that is the product of terms for all the edges of the cycle.

The edge zeta function is a product over all the cycles of these monomials inverted:  .

The edge function can therefore be evaluated at every point of the fundamental polytope by setting the appropriately. Previous work by Koetter, Li Vontobel and Walker established that each monomial in the Taylor expansion of this function corresponds to an (unscaled) pseudo-codeword of the cycle code. In particular the powers appearing for each edge (=variable of the code) correspond to re-scaled pseudo-codeword coordinates. As far as I understood, this paper shows that the coefficients of the monomials associated with a given pseudo-codeword have a meaning: their asymptotic growth rate equals to the directional derivative of the (induced) Bethe entropy in the direction of that pseudo-codeword.

I was wondering on the connections to the self-avoiding walks of Dror Weitz [Counting independent sets up to the tree threshold] (also Local approximate inference algorithms)  and the connections of the random walks to the weights of the Arora,Daskalakis, Steurer Message Passing algorithms and improved LP decoding paper.