Fig. 4

Robust exponential storage in networks of McCulloch–Pitts neurons. Error-correction performance of Hopfield networks storing all 64-cliques in \(v=128\) vertex graphs using a fully connected 8128-bit network minimizing probability flow (5) on \(50\text{,}000\) random 64-cliques (light gray line), a sparsely connected \((x, 0, 1)\) network with large deviation setting \(x = \frac{3+2p}{4k(1 + 2p)}\) and \(p=1/4\) (gray), or a sparsely connected MPF theoretical optimum (7) (black). Over 10 trials, 100 64-cliques chosen uniformly at random were p-corrupted for different p and then dynamics were converged initialized at noisy cliques. The plot shows the fraction of cliques completely recovered vs. pattern corruption p (standard deviation error bars). Dotted lines are average number of bits in a pattern retrieved correctly after converging network dynamics