  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.
