Asymptotics for Reinforced Stochastic Processes on Hierarchical Networks
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arXiv
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2025
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| author | Yang, Li Jiang, Dandan Hu, Jiang Bai, Zhidong |
| author_facet | Yang, Li Jiang, Dandan Hu, Jiang Bai, Zhidong |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_04562 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotics for Reinforced Stochastic Processes on Hierarchical Networks Yang, Li Jiang, Dandan Hu, Jiang Bai, Zhidong Statistics Theory 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. |
| title | Asymptotics for Reinforced Stochastic Processes on Hierarchical Networks |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2511.04562 |