  Scale-free networks are a recently developed approach to model the
interactions found in complex natural and man-made systems. Such networks
exhibit a power-law distribution of node link (degree) frequencies n(k) in
which a small number of highly connected nodes predominate over a much greater
number of sparsely connected ones. In contrast, in an Erdos-Renyi network each
of N sites is connected to every site with a low probability p (of the orde r
of 1/N). Then the number k of neighbors will fluctuate according to a Poisson
distribution. One can instead assume that each site selects exactly k neighbors
among the other sites. Here we compare in both cases the usual network with the
directed network, when site A selects site B as a neighbor, and then B
influences A but A does not influence B. As we change from undirected to
directed scale-free networks, the spontaneous magnetization vanishes after an
equilibration time following an Arrhenius law, while the directed ER networks
have a positive Curie temperature.
