On the number of equilibrium states in weakly coupled random networks

A fully interconnected network consisting of n elements with outputs x={xi,xi=+1,-1, 1[less-than-or-equals, slant]i[less-than-or-equals, slant]n}, connection weights wij=wijs+cwija composed of symmetric and antisymmetric parts, and dynamics described by x|->{sign([summation operator]wijxj)} is considered. Here the {wijs,wija,wkk, i<j} are i.i.d. and wijs=wjis, -wija=wjia, c is a constant in [0,[infinity]]. c=[infinity] is interpreted as wij=wija. The asymptotic behavior of the expected number of the equilibrium states of the network is studied as n-->[infinity].