Article,
Structure of S-shaped growth in innovation diffusion
Physical Review E, (May 2012)
DOI: 10.1103/physreve.85.056121
Abstract
A basic question on innovation diffusion is why the growth curve of the adopter population in a large society is often S shaped. From macroscopic, microscopic, and mesoscopic viewpoints, the growth of the adopter population is observed as the growth curve, individual adoptions, and differences among individual adoptions, respectively. The S shape can be explained if an empirical model of the growth curve can be deduced from models of microscopic and mesoscopic structures. However, even the structure of growth curve has not been revealed yet because long-term extrapolations by proposed models of S-shaped curves are unstable and it has been very difficult to predict the long-term growth and final adopter population. This paper studies the S-shaped growth from the viewpoint of social regularities. Simple methods to analyze power laws enable us to extract the structure of the growth curve directly from the growth data of recent basic telecommunication services. This empirical model of growth curve is singular at the inflection point and a logarithmic function of time after this point, which explains the unstable extrapolations obtained using previously proposed models and the difficulty in predicting the final adopter population. Because the empirical S curve can be expressed in terms of two power laws of the regularity found in social performances of individuals, we propose the hypothesis that the S shape represents the heterogeneity of the adopter population, and the heterogeneity parameter is distributed under the regularity in social performances of individuals. This hypothesis is so powerful as to yield models of microscopic and mesoscopic structures. In the microscopic model, each potential adopter adopts the innovation when the information accumulated by the learning about the innovation exceeds a threshold. The accumulation rate of information is heterogeneous among the adopter population, whereas the threshold is a constant, which is the opposite of previously proposed models. In the mesoscopic model, flows of innovation information incoming to individuals are organized as dimorphic and partially clustered. These microscopic and mesoscopic models yield the empirical model of the S curve and explain the S shape as representing the regularities of information flows generated through a social self-organization. To demonstrate the validity and importance of the hypothesis, the models of three level structures are applied to reveal the mechanism determining and differentiating diffusion speeds. The empirical model of S curves implies that the coefficient of variation of the flow rates determines the diffusion speed for later adopters. Based on this property, a model describing the inside of information flow clusters can be given, which provides a formula interconnecting the diffusion speed, cluster populations, and a network topological parameter of the flow clusters. For two recent basic telecommunication services in Japan, the formula represents the variety of speeds in different areas and enables us to explain speed gaps between urban and rural areas and between the two services. Furthermore, the formula provides a method to estimate the final adopter population.
