Title: Modeling Product Ecosystems Show Chinese title

Title: 建模产品生态系统

Abstract: This paper develops a dynamical-systems framework for modeling influence propagation in product adoption networks, formulated as a positive linear system with Metzler interaction matrices and utility-based decay. Exact solutions are derived for constant, piecewise-constant, and fully time-varying interaction structures using matrix exponentials and the Peano--Baker series. It establishes five results: (i) positive interactions guarantee nonnegative amplification, (ii) perceived utility saturates after $\approx\!3$ complementary additions (Weber--Fechner), (iii) frequency of comparable introductions dominates incremental quality improvements, (iv) reinforcing interactions yields monotone gains while decay control gives ambiguous effects, and (v) long-run retention under SIS-type dynamics is bounded by the inverse spectral radius of the adoption graph. These results extend epidemic-threshold theory and positive-systems analysis to networked adoption, yielding explicit, calibratable expressions for influence dynamics on networks. Show Chinese abstract

Abstract: 本文开发了一个动态系统框架，用于建模产品采用网络中的影响传播，其形式为具有Metzler交互矩阵和基于效用的衰减的正线性系统。 利用矩阵指数和Peano--Baker级数，推导了常数、分段常数和完全时变交互结构的精确解。 它建立了五个结果：(i) 正交互保证非负放大，(ii) 在$\approx\!3$个互补增加后感知效用趋于饱和（韦伯-费希纳），(iii) 可比较引入的频率主导增量质量改进，(iv) 强化交互产生单调收益，而衰减控制则产生模糊效果，(v) 在SIS型动力学下的长期保留受采用图的逆谱半径限制。 这些结果将流行病阈值理论和正系统分析扩展到网络化采用，为网络上的影响动态提供了明确且可校准的表达式。