On the complexity of computing equilibria for nonsymmetric analog networks

The authors ask what is the complexity of computing equilibria for physically realizable analog networks like those of J.J. Hopfield (1984) and T.J. Sejnowski (1981) with arbitrary connectivity. It is shown that, if the amplifiers are piecewise-linear, then such networks are instances of a game-theoretic model known as polymatrix games. Equilibria for the latter may be computed by vertex pivoting algorithms similar to the simplex method for linear programming, which are in practice of low order polynomial complexity. These algorithms appear to be the only ones both guaranteed to work and which are polynomial in practice. These results suggest that networks with few nonstable equilibria would be computationally attractive