Hopfield Network — Attractor Capacity

N=100 neurons, Hebbian learning — retrieve stored patterns, observe capacity cliff at P/N ≈ 0.14

P/N ratio: 0.05 Overlap m: -- Energy: -- Capacity cliff: ~0.14N
About: The Hopfield network (1982) stores P binary patterns ξ_μ ∈ {±1}^N using Hebbian weights W = (1/N)Σ_μ ξ_μ ξ_μᵀ. Retrieval works via σ_i → sgn(Σⱼ Wᵢⱼ σⱼ) until convergence. The critical capacity is P_c ≈ 0.138N (Amit-Gutfreund-Sompolinsky 1985); above this the network fails to retrieve stored patterns — spurious attractors dominate. Right plot shows retrieval overlap vs P/N.