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.

翻译：社会中的思想和语言传播通常由S形模型（如逻辑曲线）描述。然而，亚指数增长——一种在流行病学中已知的慢于指数的增长模式——在更广泛的社会现象中很大程度上被忽视了。本文提出了一种分段幂律模型，用少量参数刻画复杂的增长曲线。我们系统分析了约十亿篇与维基百科词汇关联的日语博客文章的大规模数据集，并在网络搜索趋势数据（英语、西班牙语和日语）中观察到一致的模式。对2,965个选定项目的分析表明，约55%（1,625个项目）未出现突变，且可由一个或两个分段很好地拟合。对于单分段曲线，我们发现：（i）形状参数α的众数接近0.5，表明亚指数增长普遍存在；（ii）最终传播规模主要由增长率R决定，α或持续时间T的影响较小；（iii）α随主题性质呈现变化趋势，小众/本地主题的α值较小，广泛共享主题的α值较大。此外，一个区分外向陌生人接触与内向社区互动的微观行为模型表明，α可解释为外向型沟通偏好的指标。这些发现表明，亚指数增长是社会传播的常见模式，我们的模型为一致地描述、比较和解释复杂多样的增长曲线提供了实用框架。