Number of Local Attractors of Scale-Free Networks of Hopfield

Abstract

Estimates of the number of local attractors for the Hopfield model of attractor neural network with continuous time and states where the neuron interaction graph has a scale-free structure are considered. The number of local attractors defines the network capacity, which is an important network characteristic. Numerous works were devoted to the problem of capacity estimations but mainly Boolean networks and the Hopfield models with symmetric interactions were studied.

In the second case the capacity is proportional to the neuron number N. An estimation of the capacity via characteristics of the network interaction graph is found. This estimate implies that the capacity may increase as  exp( cN^a ), where c, a > 0. Furthermore, a formula, which connects the capacity and the number of strongly connected neurons (hubs) in the network has been found by computer simulations. We show that the logarithm of the capacity is proportional to the hub number and the hub number is proportional to the root of N.

Results can have applications to associative memory neural models and morphogenesis modeling by genetic networks.