Modeling Product Ecosystems

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.