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The non-backtracking operator was recently shown to give a redemption forspectral clustering in sparse graphs. In this paper we considernon-backtracking operator for Ising model on a general graph with a generalcoupling distribution by linearizing Belief Propagation algorithm atparamagnetic fixed-point. The spectrum of the operator is studied, the sharpedge of bulk and possible real eigenvalues outside the bulk are computedanalytically as a function of couplings and temperature. We show theapplications of the operator in attractor neural networks. At thermodynamiclimit, our result recovers the phase boundaries of Hopfield model obtained byreplica method. On single instances of Hopfield model, its eigenvectors can beused to retrieve all patterns simultaneously. We also give an example on how tocontrol the neural networks, i.e. making network more sparse while keepingpatterns stable, using the non-backtracking operator and matrix perturbationtheory.

Non-backtracking operator for Ising model and its application in attractor neural networks