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Modeling Product Ecosystems

Tridib Banerjee

TL;DR

The paper addresses influence propagation in product-ecosystem networks by formulating a positive linear system with Metzler interaction matrices and utility-based decay: $\dot{\boldsymbol{\alpha}}(t)=M(t)\boldsymbol{\alpha}(t)+\boldsymbol{u}(t)$. It derives exact solutions for constant, piecewise-constant, and fully time-varying interactions using matrix exponentials and the Peano–Baker series, enabling precise analysis of dynamic amplification and saturation. Key contributions include establishing positivity and stability properties, a Weber–Fechner saturation law for perceived utility, a frequency-dependent growth interpretation, a robust ROI-guided investment framework, and a spectral-radius bound on long-run retention in SIS-like diffusion. The framework yields explicit, calibratable expressions linking marketing inputs, inter-product interactions, and social diffusion to practical design and investment decisions in 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.

Modeling Product Ecosystems

TL;DR

The paper addresses influence propagation in product-ecosystem networks by formulating a positive linear system with Metzler interaction matrices and utility-based decay: . It derives exact solutions for constant, piecewise-constant, and fully time-varying interactions using matrix exponentials and the Peano–Baker series, enabling precise analysis of dynamic amplification and saturation. Key contributions include establishing positivity and stability properties, a Weber–Fechner saturation law for perceived utility, a frequency-dependent growth interpretation, a robust ROI-guided investment framework, and a spectral-radius bound on long-run retention in SIS-like diffusion. The framework yields explicit, calibratable expressions linking marketing inputs, inter-product interactions, and social diffusion to practical design and investment decisions in 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 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.

Paper Structure

This paper contains 16 sections, 5 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Theory
  3. Properties: Positivity, Stability, Boundedness
  4. Exact Solutions
  5. Scalar warm-up (intuition)
  6. Constant $M$ (matrix integrating factor)
  7. Piecewise-constant $M$ (interval recursion)
  8. Fully time-varying $M(t)$ (state-transition matrix)
  9. Discrete-Time Form and Estimation
  10. Propositions
  11. Amplification under Constant Interactions
  12. Steady Identical Growth and Weber--Fechner Saturation
  13. Unsteady Growth -- Frequency vs. Quality
  14. Optimal Investment -- Interactions vs. Cost Cuts
  15. Retention Bounded by Spectral Radius
  16. ...and 1 more sections

Key Result

Proposition 1

If $\Lambda$ is constant with $\Lambda_{ij}\ge 0$ for $i\neq j$, then for all $t\ge t_0$, $\alpha_i(t)\;\ge\;\alpha_i^\circ(t)\quad\text{and}\quad\mathcal{A}_i(t)\ge 1$ whenever $\alpha_i^\circ(t)>0$.

Theorems & Definitions (17)

  • Remark 1
  • Proposition 1
  • proof
  • Remark 2
  • Proposition 2
  • proof
  • Remark 3
  • Proposition 3
  • proof
  • Proposition 4
  • ...and 7 more