Higher order neural networks for combinatorial optimisation improving the scaling properties of the Hopfield network

The dynamics of the Hopfield network are investigated to determine why the network does not scale well to large problem sizes. It is seen that the Hopfield network encourages the formation of locally optimal segments, resulting in multiple seed points. The evolution of the system state from such seed points and the segmentation of the final solution are readily visualised on a simple graph 2-colouring problem (Ising Spin). By extending the neighbourhood in which the formation of locally optimal segments is encouraged, higher order neural networks (HONNs) are developed as a means to improve the scaling properties of the Hopfield network