Attractor Degeneracy: A Theoretical Analysis of Equivalent Memory Classes in Hopfield Neural Networks

Hopfield neural networks (HNNs) realize associative memory through dynamics induced by memories. Different memory matrices can nevertheless generate conjugate network dynamics, a phenomenon we call attractor degeneracy. We study the equivalence relation generated by row and column permutations and sign flips in binary HNNs. With synchronous updates and spin retention at zero field, equivalent matrices have update maps related by a signed coordinate relabeling. This bijection preserves trajectories, periodic attractors, basin sizes, and Hamming distances. We describe the equivalence classes as group orbits, derive an exact count among all $2^{MN}$ binary memory matrices with $M$ memories and $N$ neurons, and give an orbit-recognition method that recovers the row and column transformations. The count avoids enumerating the memory space, while a recovered transformation allows the dynamics of one representative to be transferred to any equivalent memory matrix. Exact complete-space and independently sampled candidate-bank experiments preserve the reported statistics while reducing repeated HNN evaluations beyond a column-only quotient.