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

Abstract: The non-backtracking operator was recently shown to give a redemption for spectral clustering in sparse graphs. In this paper we consider non-backtracking operator for Ising model on a general graph with a general coupling distribution by linearizing Belief Propagation algorithm at paramagnetic fixed-point. The spectrum of the operator is studied, the sharp edge of bulk and possible real eigenvalues outside the bulk are computed analytically as a function of couplings and temperature. We show the applications of the operator in attractor neural networks. At thermodynamic limit, our result recovers the phase boundaries of Hopfield model obtained by replica method. On single instances of Hopfield model, its eigenvectors can be used to retrieve all patterns simultaneously. We also give an example on how to control the neural networks, i.e. making network more sparse while keeping patterns stable, using the non-backtracking operator and matrix perturbation theory.