Untitled

Abstract

The parallel dynamics of extremely diluted symmetric Q-Ising neural networks is studied for arbitrary Q using a probabilistic approach. In spite of the extremely diluted architecture the feedback correlations arising from the symmetry prevent a closed-form solution in contrast with the extremely diluted asymmetric model. A recursive scheme is found determining the complete time evolution of the order parameters taking into account all feedback. It is based upon the evolution of the distribution of the local field, as in the fully connected model. As an illustrative example an explicit analysis is carried out for the Q = 2 and Q = 3 model. These results agree with and extend the partial results existing for Q = 2. For Q > 2 the analysis is entirely new. Finally, equilibrium fixed-point equations are derived and a capacity-gain function diagram is obtained.

Thermodynamic properties of extremely diluted symmetricQ-Ising neural networks

“…At this point a couple of remarks are in order. First, we note that the fixed-point equations and, hence, also the α − b phase diagram are precisely the same as those obtained from an exact dynamical approach (see [1] eqs. ( 23)-(25) and fig.…”

“…Recently the dynamics of extremely diluted symmetric Q-Ising neural networks has been solved completely [1]. In spite of the extremely diluted architecture, precisely the symmetry causes feedback correlations from the second time step onwards, in contrast with an asymmetric architecture [2,3], complicating the dynamics in a nontrivial way.…”

Parallel dynamics of the asymmetric extremely diluted Ashkin-Teller neural network

“…These correlations are caused by feedback loops and common ancestors. In contrast to the Hopfield model, where the dynamics has been solved taking into account all the correlations [10,11], the presence of two types of spins makes the analysis of the correlations in the atnn very complicated. The underlying reason is the existence of two sources of correlations.…”

Parallel dynamics of the fully connected Blume–Emery–Griffiths neural network

“…This microscopic dependence gives rise to a macroscopic contribution after summing and taking the limit N → ∞. Therefore, we follow a procedure similar to the one used in the Q-Ising model [4], [10] by isolating in the local fields precisely the contributions arising from these dependences.…”