Hopfield Model on Incomplete Graphs

Abstract: We consider the Hopfield model on graphs. Specifically we compare five different incomplete graphs on 4 or 5 vertices’s including a cycle, a path and a star. Provided is a proof of the Hamiltonian being monotonically decreasing under asynchronous network dynamics. This result is applied to the treated incomplete graphs to derive exact values for the incre- mental drop in energy on pattern sizes 2, 4, and an arbitrary m under restriction. Special cases provided includes evaluating the network on a graph as a union of two independent components, and additionally one example using a deterministic dilute variable. Furthermore we study the stability of patterns considering a Hopfield model with synchronous net- work dynamics for two different incomplete graphs using simulations.