Retrieval behavior and thermodynamic properties of symmetrically dilutedQ-Ising neural networks

Abstract

The retrieval behavior and thermodynamic properties of symmetrically diluted Q-Ising neural networks are derived and studied in replica-symmetric mean-field theory generalizing earlier works on either the fully connected or the symmetrical extremely diluted network. Capacity-gain parameter phase diagrams are obtained for the Q = 3, Q = 4 and Q = ∞ state networks with uniformly distributed patterns of low activity in order to search for the effects of a gradual dilution of the synapses. It is shown that enlarged regions of continuous changeover into a region of optimal performance are obtained for finite stochastic noise and small but finite connectivity. The de Almeida-Thouless lines of stability are obtained for arbitrary connectivity, and the resulting phase diagrams are used to draw conclusions on the behavior of symmetrically diluted networks with other pattern distributions of either high or low activity.

The three-state layered neural network with finite dilution

“…In view of a similar recent result for a three-state recurrent diluted network with symmetric synaptic connections [3], one may conclude that this is a feature of finite dilution which is independent of both the network architecture and of the interaction symmetry. Thus, provided there is an above minimum threshold such that a network attains the ability to retrieve a nominated pattern after eliminating undesirable transient states, the good retrieval performance does not depend in an essential way on a precise threshold adjustment and this may explain why biological networks, in which there are no precise thresholds, can have a good performance despite a fraction of missing synaptic connections.…”

Synchronous versus sequential updating in the three-state Ising neural network with variable dilution

“…In order to calculate the free energy we use the replica method as applied to dilute systems [13], [14], [24]- [27]. Since this method is really standard by now, at least for sequential updating, we refrain from giving any detailed calculations but concentrate on the main results and the differences between sequential and synchronous updating.…”

“…The latter studies make use of the so-called signal-to-noise analysis (see, e.g., [1] for references on this method). An extension to the whole dilution range, in analogy with [14] has not yet been given.…”

The Blume-Emery-Griffiths neural network with synchronous updating and variable dilution

“…We adapt the standard replica technique as applied to dilute models [12]- [16] to synchronous updating in order to calculate the free energy of the model. Since the method is rather standard by now, although the specific details are rather involved, we only indicate the main steps.…”