A Class of Automata Networks for Diffusion of Innovations Driven by Riccati Equations

Innovation diffusion processes are generally described at aggregate level with models like the Bass model (1969) and the Generalized Bass Model (1994). However, the recognized importance of communication channels between agents has recently suggested the use of agent-based models, like Cellular Automata. We argue that an adoption process is nested in a communication network that evolves dynamically and implicitly generates a non-constant potential market. Using Cellular Automata we propose a two{phase model of an innovation diffusion process. First we describe the Communication Network necessary for the awareness of an innovation. Then, we model a nested process representing the proper adoption dynamics. Through a "Mean Field Approximation" we propose a continuous representation of the discrete time equations derived by our Automata Network. This constitutes a special non autonomous Riccati equation, not yet described in well-known international catalogues. The main results refer to the closed form solution of this equation and to the corresponding statistical analysis for identfication and inference. We discuss an application in the field of bank services.i