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This article unveils a new relation between the Nishimori temperatureparametrizing a distribution P and the Bethe free energy on random Erdos-Renyigraphs with edge weights distributed according to P. Estimating the Nishimoritemperature being a task of major importance in Bayesian inference problems, asa practical corollary of this new relation, a numerical method is proposed toaccurately estimate the Nishimori temperature from the eigenvalues of the BetheHessian matrix of the weighted graph. The algorithm, in turn, is used topropose a new spectral method for node classification in weighted (possiblysparse) graphs. The superiority of the method over competing state-of-the-artapproaches is demonstrated both through theoretical arguments and real-worlddata experiments.

Nishimori meets Bethe: a spectral method for node classification in sparse weighted graphs