  This article unveils a new relation between the Nishimori temperature
parametrizing a distribution P and the Bethe free energy on random Erdos-Renyi
graphs with edge weights distributed according to P. Estimating the Nishimori
temperature being a task of major importance in Bayesian inference problems, as
a practical corollary of this new relation, a numerical method is proposed to
accurately estimate the Nishimori temperature from the eigenvalues of the Bethe
Hessian matrix of the weighted graph. The algorithm, in turn, is used to
propose a new spectral method for node classification in weighted (possibly
sparse) graphs. The superiority of the method over competing state-of-the-art
approaches is demonstrated both through theoretical arguments and real-world
data experiments.
