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 2,963 items, selected for reliable estimation (e.g., sufficient duration/peak, monotonic growth), reveals that 1,625 (55%) diffusion patterns without abrupt level shifts were adequately described by one or two segments. For single-segment curves, we found that (i) the mode of the shape parameter $α$ was near 0.5, indicating prevalent sub-exponential growth; (ii) the peak diffusion scale is primarily determined by the growth rate $R$, with minor contributions from $α$ or the duration $T$; and (iii) $α$ 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 of outward (stranger) vs. inward (community) contact suggests that $α$ 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扩散传统上采用S形模型（如逻辑斯蒂曲线）描述。然而，亚指数增长——流行病学中已知的一种慢于指数增长的模式——在更广泛的社会现象中很大程度上被忽视了。本文提出一个分段幂律模型，利用少量参数刻画复杂增长曲线。我们系统分析了约十亿篇日本博客文章（与维基百科词汇关联）的大规模数据集，并在网络搜索趋势数据（英语、西班牙语、日语）中观察到一致模式。对2963个符合可靠估计条件（如充分时长/峰值、单调增长）项目的分析显示，1625个（55%）无明显水平偏移的扩散模式可由一个或两个分段充分描述。对于单段曲线，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）峰值扩散规模主要由增长率R决定，α或持续时间T的贡献较小；（iii）α随主题性质呈变化趋势：利基/局部主题的α值较小，而广泛共享主题的α值较大。此外，基于向外（陌生人）与向内（社群）接触的微观行为模型表明，α可解读为对外向沟通偏好的指数。这些发现表明亚指数增长是社会扩散的常见模式，我们的模型为一致性描述、比较和解读复杂多样的增长曲线提供了实用框架。