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由S形模型（如逻辑斯蒂曲线）描述。然而，次指数增长——一种在流行病学中已知的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。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关联维基百科词汇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。我们对2963个条目（基于可靠估计条件选取，如足够的持续时间/峰值、单调增长）的分析表明，1625个（55%）无突变水平偏移的扩散模式可由一个或两个分段充分描述。对于单分段曲线，我们发现：（i）形状参数$α$的众数接近0.5，表明次指数增长普遍存在；（ii）峰值扩散规模主要由增长率$R$决定，$α$或持续时间$T$贡献较小；（iii）$α$呈现随话题性质变化的趋势，对于小众/局部话题较小，对于广泛共享话题较大。此外，一个关于外向（陌生人）与内向（社群）接触的微观行为模型表明，$α$可被解释为对外向型交流偏好程度的指标。这些发现表明次指数增长是社会扩散的常见模式，本模型为一致描述、比较和解读复杂多样的增长曲线提供了实用框架。