J. Phys. I France
Volume 2, Numéro 9, September 1992
Page(s) 1791 - 1801
DOI: 10.1051/jp1:1992245
J. Phys. I France 2 (1992) 1791-1801

Phase diagrams and the instability of the spin glass states for the diluted Hopfield neural network model

Andrew Canning and Jean-Pierre Naef

Département de Physique Théorique, Université de Genève, CH-1211 Genève 4, Switzerland

(Received 17 February 1992, accepted in final form 7 May 1992)

The replica-symmetric order parameter equations derived in [2, 4] for the symmetrically diluted Hopfield neural network model [1] are solved for different degrees of dilution. The dilution is random but symmetric. Phase diagrams are presented for c=1, 0.1, 0.001 and $c\to 0$, where c is the fractional connectivity. The line $T_{\rm c}$ where the memory states become global minima (having lower free energy than the spin glass states) is also found for different values of c. It is found that the effect of dilution is to destabilize the spin glass states and the line $T_{\rm c}$ is driven rapidly towards the line $T_{\rm M}$; the phase transition line where the memory states first stabilize. All the results are derived in the context of replica symmetry so are expected to be incorrect in certain parts of the phase diagram (the error increasing the further below the replica symmetry breaking lines we are) but our results do suggest in a general way that dilution, even in small quantities, increases the stability of the memory states with respect to the spin glass states.

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