Guaranteed convergence in a class of Hopfield networks.

A class of symmetric Hopfield networks with nonpositive synapses and zero threshold is analyzed in detail. It is shown that all stationary points have a one-to-one correspondence with the minimal vertex covers of certain undirected graphs, that the sequential Hopfield algorithm as applied to this class of networks converges in at most 2n steps (n being the number of neurons), and that the parallel Hopfield algorithm either converges in one step or enters a two-cycle in one step. The necessary and sufficient condition on the initial iterate for the parallel algorithm to converge in one step are given. A modified parallel algorithm which is guaranteed to converge in [3n/2] steps ([x] being the integer part of x) for an n-neuron network of this particular class is also given. By way of application, it is shown that this class naturally solves the vertex cover problem. Simulations confirm that the solution provided by this method is better than those provided by other known methods.