The capacity of associative memories with malfunctioning neurons

Hopfield associative memories with αn malfunctioning neurons are considered. Using some facts from exchangeable events theory, the asymptotic storage capacity of such a network is derived as a function of the parameter α under stability and attractivity requirements. It is shown that the asymptotic storage capacity is (1-α)2n/(4 log n) under stability and (1-α)²(1-2ρ)²n/(4 log n) under attractivity requirements, respectively. Comparing these capacities with their maximum values corresponding to the case when there is no malfunctioning neurons, α=0, shows the robustness of the retrieval mechanism of Hopfield associative memories with respect to the existence of malfunctioning neurons. This result also supports the claim that neural networks are fault tolerant