Stability analysis of higher-order neural networks for combinatorial optimization

Recurrent neural networks with higher order connections, from here on referred to as higher-order neural networks (HONNs), may be used for the solution of combinatorial optimization problems. In Ref. 5 a mapping of the traveling salesman problem (TSP) onto a HONN of arbitrary order was developed, thereby creating a family of related networks that can be used to solve the TSP. In this paper, we explore the trade-off between network complexity and quality of solution that is made available by the HONN mapping of the TSP. The trade-off is investigated by undertaking an analysis of the stability of valid solutions to the TSP in a HONN of arbitrary order. The techniques used to perform the stability analysis are not new, but have been widely used elsewhere in the literature. The original contribution in this paper is the application of these techniques to a HONN of arbitrary order used to solve the TSP. The results of the stability analysis show that the quality of solution is improved by increasing the network complexity, as measured by the order of the network. Furthermore, it is shown that the Hopfield network, as the simplest network in the family of higher-order networks, is expected to produce the poorest quality of solution.