Efficient Hopfield pattern recognition on a scale-free neural network
  Neural networks are supposed to recognise blurred images (or patterns) of $N$ pixels (bits) each. Application of the network to an initial blurred version of one of $P$ pre-assigned patterns should converge to the correct pattern. In the "standard" Hopfield model, the $N$ "neurons'' are connected to each other via $N^2$ bonds which contain the information on the stored patterns. Thus computer time and memory in general grow with $N^2$. The Hebb rule assigns synaptic coupling strengths proportional to the overlap of the stored patterns at the two coupled neurons. Here we simulate the Hopfield model on the Barabasi-Albert scale-free network, in which each newly added neuron is connected to only $m$ other neurons, and at the end the number of neurons with $q$ neighbours decays as $1/q^3$. Although the quality of retrieval decreases for small $m$, we find good associative memory for $1 \ll m \ll N$. Hence, these networks gain a factor $N/m \gg 1$ in the computer memory and time. 
Dietrich Stauffer, Amnon Aharony, Luciano da Fontoura Costa, and Joan   Adler