Sub-exponential Growth in Online Word Usage: A Piecewise Power-Law Model

The diffusion of ideas and language in society has conventionally been described by S-shaped models, such as the logistic curve. However, the role of sub-exponential growth -a slower than exponential pattern known in epidemiology- has been largely overlooked in broader social phenomena. Here, we present a piecewise power-law model to characterize complex growth curves with a few parameters. We systematically analyzed a large-scale dataset of approximately one billion Japanese blog articles linked to Wikipedia vocabulary, and observed consistent patterns in web search trend data (English, Spanish, and Japanese). Our analysis of the 2,965 selected items reveals that about 55% (1,625 items) were found to have no abrupt jumps and were well captured by one or two segments. For single-segment curves, we found that (i) the mode of the shape parameter alpha was near 0.5, indicating prevalent sub-exponential growth; (ii) the ultimate diffusion scale is primarily determined by the growth rate R, with minor contributions from alpha or the duration T; and (iii) alpha showed a tendency to vary with the nature of the topic, being smaller for niche/local topics and larger for widely shared ones. Furthermore, a micro-behavioral model distinguishing outward contact with strangers from inward interaction within their community suggests that alpha can be interpreted as an index of the preference for outward-oriented communication. These findings suggest that sub-exponential growth is a common pattern of social diffusion, and our model provides a practical framework for consistently describing, comparing, and interpreting complex and diverse growth curves.