On the performance of Hopfield network for graph search problem

This paper presents a study on the performance of the Hopfield neural network algorithm for the graph path search problem. Specifically, performance of the Hopfield network is studied from the dynamic systems stability perspective. Simulations of the time behavior of the neural network is augmented with exhaustive stability analysis of the equilibrium points of the network dynamics. The goal is to understand the reasons for the well-known deficiency of the Hopfield network algorithm: the inability to scale up with the problem size. A recent procedure, which establishes solutions as stable equilibrium points in the state space of the network dynamics, is employed to define the constraint weight parameters of the Hopfield neural network. Simulation study of the network and stability analysis of equilibrium points indicate that a large set of nonsolution equilibrium points also become stable whenever constraint weight parameters are set to make the solution equilibrium points stable. As the problem size grows, the number of stable non-solution equilibrium points increases at a much faster rate than the number of stable solution equilibrium points: the network becomes more likely to converge to a non-solution equilibrium point.