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Central limit theorems for interacting innovation processes, related statistical tools and general results

Giacomo Aletti, Irene Crimaldi, Andrea Ghiglietti

TL;DR

This work analyzes a model of interacting urns, where the probability of generating or reusing elements in one process is influenced by the histories of others, and develops statistical tools to infer the structure and strength of inter-process influence.

Abstract

We study a networked system of innovation processes, where each process is modeled as an urn with infinitely many colors-a classical framework for capturing the emergence of novelties. Extending this paradigm, we analyze a model of interacting urns, where the probability of generating or reusing elements in one process is influenced by the histories of others. This interaction is governed by two matrices that control innovation triggering and reinforcement dynamics across the system. The core contribution of this work is a detailed analysis of the second-order asymptotic behavior of the model. Building on these theoretical results, we develop statistical tools to infer the structure and strength of inter-process influence. The methodology is framed in a general setting, making it broadly applicable. We validate our approach with applications to two real-world datasets from Reddit discussions and Gutenberg text corpora.

Central limit theorems for interacting innovation processes, related statistical tools and general results

TL;DR

This work analyzes a model of interacting urns, where the probability of generating or reusing elements in one process is influenced by the histories of others, and develops statistical tools to infer the structure and strength of inter-process influence.

Abstract

We study a networked system of innovation processes, where each process is modeled as an urn with infinitely many colors-a classical framework for capturing the emergence of novelties. Extending this paradigm, we analyze a model of interacting urns, where the probability of generating or reusing elements in one process is influenced by the histories of others. This interaction is governed by two matrices that control innovation triggering and reinforcement dynamics across the system. The core contribution of this work is a detailed analysis of the second-order asymptotic behavior of the model. Building on these theoretical results, we develop statistical tools to infer the structure and strength of inter-process influence. The methodology is framed in a general setting, making it broadly applicable. We validate our approach with applications to two real-world datasets from Reddit discussions and Gutenberg text corpora.
Paper Structure (28 sections, 16 theorems, 162 equations, 2 figures, 2 tables)

This paper contains 28 sections, 16 theorems, 162 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related literature
  3. Structure of the work
  4. Model and first-order convergence results
  5. Central limit theorems
  6. Statistical inference
  7. The case $N = 2$
  8. Hypothesis test on $\Gamma$
  9. Hypothesis test on $W$
  10. Confidence interval for $\widetilde{P}_\infty(c)$
  11. Mean-field interaction for $N\geq 2$
  12. Hypothesis test on $\Gamma$
  13. Hypothesis test on $W$
  14. Confidence interval for $\widetilde{P}_\infty(c)$
  15. Applications
  16. ...and 13 more sections

Key Result

Theorem 2.1

Suppose that the matrix $\Gamma=(\gamma_{j,h})_{j,h=1,\dots,N}$ is irreducible. Denote by $\gamma^*\in (0,1)$ the Perron-Frobenius eigenvalue of $\Gamma$ and by ${\mathbf{u}}$ its left eigenvector. Then, we have where ${\mathbf{D}} ^*_t = (D^*_{t,1},\ldots, D^*_{t,N})^\top$ and $D^{**}_{\infty}$ is an integrable strictly positive random variable.

Figures (2)

  • Figure 1: Confidence interval (see \ref{['eq:confidence_interval_N=2']}) for the random variable $\widetilde{P}_\infty(c)$ at time-step $t=10^3$ and related to item $c$ with highest ${\boldsymbol{1}}^\top {\mathbf{K}} _t(c)$ in the case $N=2$, $\Gamma$ defined as in \ref{['eq:elements_Gamma_N_equal_2']} with $r=0.75$, $\gamma^{*}=0.75$, $\eta=1/2$ and $\iota_{\Gamma}=1$, and $W$ defined as in \ref{['eq:elements_W_N_equal_2']} with $\eta=1/2$ and $\iota_W=1.25$. Number of simulations $S=200$ (sorted on the $x$-axis according to the value of ${\boldsymbol{1}}^\top {\mathbf{K}} _t(c)$). The crosses represent the value (reported on the $y$-axis) of $\widetilde{P}_\infty(c)$ estimated by ${\boldsymbol{1}}^\top {\mathbf{K}} _{t_\infty}(c)/(Nt_\infty)$ at time-step $t_{\infty}=10^5$.
  • Figure 2: Confidence interval (see \ref{['eq:confidence_interval_mean_field_case']}) for the random variable $\widetilde{P}_\infty(c)$ at time-step $t=10^3$ and related to item $c$ with highest ${\boldsymbol{1}}^\top {\mathbf{K}} _t(c)$ in the case $N=3$, $\Gamma$ defined as in \ref{['eq:elements_Gamma_mean-field']} with $\phi=0.75$ and $\iota_{\Gamma}=0.8$, and $W$ defined as in \ref{['eq:elements_W_mean-field']} with $\iota_W=0.8$. Number of simulations $S=200$ (sorted on the $x$-axis according to the value of ${\boldsymbol{1}}^\top {\mathbf{K}} _t(c)$). The crosses represent the value (reported on the $y$-axis) of $\widetilde{P}_\infty(c)$ estimated by ${\boldsymbol{1}}^\top {\mathbf{K}} _{t_\infty}(c)/(Nt_\infty)$ at time-step $t_{\infty}=10^5$.

Theorems & Definitions (33)

  • Theorem 2.1: ale-cri-ghi-innovation-2023
  • Theorem 2.2: ale-cri-ghi-innovation-2023
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • ...and 23 more