Control-parameter scaling in a Hopfield-Tank list-matching network
Keith D. Kastella · Physical Review A · 1992
A number of authors have reported poor scaling with size N in numerical studies of the neural-network-based algorithm for the ``traveling-salesman'' problem proposed by Hopfield and Tank [Biol. Cybern. 52, 141 (1984)]. In this paper, the scaling with N of a closely related algorithm for solving bipartite list matching is analyzed in an expansion in powers of 1/lnN. A self-averaging solution is obtained by assuming row and column exchange symmetry and directly evaluating the partition function. This yields a prediction for the error rate that increases proportionally to lnN for fixed values of the control parameters, confirmed here for this model by numerical simulation. The same result can be derived using the replica method, assuming that the solutions are replica symmetric. The analysis predicts that scaling the inhibitory coupling with ${\mathit{N}}^{\mathrm{\ensuremath{\delta}}}$, \ensuremath{\delta}>0, drives the error rate asymptotically to 0 with increasing N.