Graph Degree Sequence Solely Determines the Expected Hopfield Network Pattern Stability

We analyze the effect of network topology on the pattern stability of the Hopfield neural network in the case of general graphs. The patterns are randomly selected from a uniform distribution. We start the Hopfield procedure from some pattern v. An error in an entry e of v is the situation where, if the procedure is started at e, the value of e flips. Such an entry is an instability point. Note that we disregard the value at e by the end of the procedure, as well as what happens if we start the procedure from another pattern or another entry of v. We measure the instability of the system by the expected total number of instability points of all the patterns.

Our main result is that the instability of the system does not depend on the exact topology of the underlying graph, but rather only on its degree sequence. Moreover, for a large number of nodes, the instability can be approximated by , where is the standard normal distribution function and are the degrees of the nodes.