Multiple-valued energy function in neural networks with asymmetric connections.

We apply the graphic transformation method[11],[25] to obtain the steady state distribution of asymmetric Boltzmann machines as an extension of the symmetric equilibrium case. We give the magnitude of deviation from the equilibrium explicitly as a function of the asymmetry in the connections between the neurons. We show that the steady state of asymmetric Boltzmann machines is characterized by multiple energy values, rather than by a single energy value as in the equilibrium state of symmetric Boltzmann machines. The equilibrium scalar energy function is generalized to a multiple-valued energy function in the case of asymmetric Boltzmann machines. PACS Numbers ; 87.10.+e, 02.10.+w, 42.79.Ta