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.