Morphology of technological levels in an innovation propagation model

We study the dynamical properties of the propagation of innovation on a two-dimensional lattice, random network, scale-free network, and Cayley tree. In order to investigate the diversity of technological level, we study the scaling property of width, W(N,t), which represents the root mean square of the technological level of agents. Here, N is the total number of agents. From the numerical simulations, we find that the steady-state value of W(N,t), W(sat)(N), scales as W(sat)(N) ∼ N(-1/2) when the system is in a flat ordered phase for d ≥ 2. In the flat ordered phase, most of the agents have the same technological level. On the other hand, when the system is in a smooth disordered phase, the value of W(sat)(N) does not depend on N. These behaviors are completely different from those observed on a one-dimensional (1D) lattice. By considering the effect of the underlying topology on the propagation dynamics for d ≥ 2, we also provide a mean-field analysis for W(sat)(N), which agrees very well with the observed behaviors of W(sat)(N). This directly shows that the morphological properties in order-disorder transition on a 1D lattice is completely different from that on higher dimensions. It also provides an evidence that the upper critical dimension for the roughening transition of the propagation of innovation is d(u)=2.