Asymptotics for Reinforced Stochastic Processes on Hierarchical Networks

TL;DR

This work advances the study of reinforced stochastic processes on hierarchical networks by establishing almost-sure synchronization to a random limit $Z_\infty$ whose distribution is determined by the leading subgroup, and by deriving a comprehensive joint CLT for the coupled process $({\bf Z}_n, {\bf N}_n)$. The analysis accommodates reducible and non-diagonalizable adjacency matrices with block upper-triangular structure, revealing phase transitions in convergence rates governed by the step-size exponent $\gamma$ and the Jordan block geometry of ${\bf W}$. In particular, non-diagonalizability introduces new leading covariance terms and logarithmic scalings at the critical case $\gamma=1$, linking spectral data to second-order fluctuations. Building on these results, the paper develops a practical inference framework for hierarchical networks, including hypothesis tests for network structure and confidence regions for both the synchronization limit and structural parameters, with simulations demonstrating the theoretical phenomena such as robust synchronization and the dominant influence of the leading subgroup on long-run behavior.

Abstract

In this paper, we analyze the asymptotic behavior of a system of interacting reinforced stochastic processes $({\bf Z}_n, {\bf N}_n)_n$ on a directed network of $N$ agents. The system is defined by the coupled dynamics ${\bf Z}_{n+1}=(1-r_{n}){\bf Z}_{n}+r_{n}{\bf X}_{n+1}$ and ${\bf N}_{n+1}=(1-\frac{1}{n+1}){\bf N}_n+\frac{1}{n+1}{\bf X}_{n+1}$, where agent actions $\mathbb{P}(X_{n+1,j}=1\mid{\cal F}_n)=\sum_{h} w_{hj}Z_{nh}$ are governed by a column-normalized adjacency matrix ${\bf W}$, and $r_n \sim cn^{-γ}$ with $γ\in (1/2, 1]$. Existing asymptotic theory has largely been restricted to irreducible and diagonalizable ${\bf W}$. We extend this analysis to the broader and more practical class of reducible and non-diagonalizable matrices ${\bf W}$ possessing a block upper-triangular form, which models hierarchical influence. We first establish synchronization, proving $({\bf Z}^\top_n, {\bf N}^\top_n)^\top \to Z_\infty {\bf 1}$ almost surely, where the distribution of the limit $Z_\infty$ is shown to be determined solely by the internal dynamics of the leading subgroup. Furthermore, we establish a joint central limit theorem for $({\bf Z}_n,{\bf N}_n)_n$, revealing how the spectral properties and Jordan block structure of ${\bf W}$ govern second-order fluctuations. We demonstrate that the convergence rates and the limiting covariance structure exhibit a phase transition dependent on $γ$ and the spectral properties of ${\bf W}$. Crucially, we explicitly characterize how the non-diagonalizability of ${\bf W}$ fundamentally alters the asymptotic covariance and introduces new logarithmic scaling factors in the critical case ($γ=1$). These results provide a probabilistic foundation for statistical inference on such hierarchical network structures.

Figures (2)

• Figure 1: Distribution of the synchronization limit $Z_\infty$ for the two-group hierarchical network with strong self-reinforcement ($\alpha=0.8$). The six panels correspond to different initial states for the leading ${\cal G}_1$ and subsequent ${\cal G}_2$ subgroups. Within each panel, the four overlapping density curves represent the four agents.
• Figure 2: Distribution of the synchronization limit $Z_\infty$ for the two-group hierarchical network with weak self-reinforcement ($\alpha=0.2$). The six panels correspond to different initial states for the leading ${\cal G}_1$ and subsequent ${\cal G}_2$ subgroups. Within each panel, the four overlapping density curves represent the four agents.

Theorems & Definitions (44)

• Theorem 3.1: Synchronization
• Theorem 3.2
• Corollary 3.3
• Remark 3.1
• Remark 3.2
• Theorem 3.4
• Theorem 3.5: Convergence Rate for $1/2<\gamma<1$
• Remark 3.3
• Theorem 3.6: Convergence Rate for $\gamma=1$, $\tau<1-(2c)^{-1}$
• Remark 3.4