Hopfield energy of the random neural network

Hopfield´s seminal work on energy functions for neural networks and other early work has drawn much attention to neural heuristics for combinatorial optimization. These heuristics are often very time consuming, because of the need for randomization or Monte Carlo simulation during the search for solutions. The random neural network (RNN) model has the nice property of being analytically solvable, and therefore computationally fast, so that any application of the model is based on obtaining solutions to a simple system of fixed-point equations. In this paper we first introduce the binary random network model and show that it has a Hopfield energy which it minimizes and which can be used for optimization problems. We illustrate this by the search for heuristic solutions to the minimum node covering problem (MCP) for graphs, which we also then proceed to solve using the full RNN. We then turn to a definition of Hopfield energy for the RNN model, and prove that it is minimized at each fixed-point iteration used in solving the RNN model´s equations. This is again illustrated in relation to the MCP