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 $r_n \sim cn^{-\gamma}$ with $\gamma \in (1/2, 1]$. 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 $\gamma$ 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 ($\gamma=1$). These results provide a probabilistic foundation for statistical inference on such hierarchical network structures.

翻译：本文研究了一个由N个智能体构成的有向网络上相互作用强化随机过程系统$({\\bf Z}_n, {\\bf N}_n)_n$的渐近行为。该系统由耦合动力学方程${\\bf Z}_{n+1}=(1-r_{n}){\\bf Z}_{n}+r_{n}{\\bf X}_{n+1}$和${\\bf N}_{n+1}=(1-\\frac{1}{n+1}){\\bf N}_n+\\frac{1}{n+1}{\\bf X}_{n+1}$定义，其中智能体动作概率$\\mathbb{P}(X_{n+1,j}=1\\mid{\\cal F}_n)=\\sum_{h} w_{hj}Z_{nh}$由列归一化邻接矩阵${\\bf W}$控制，且$r_n \\sim cn^{-\\gamma}$，其中$\\gamma \\in (1/2, 1]$。现有渐近理论主要局限于不可约且可对角化的${\\bf W}$矩阵。我们将分析扩展到更广泛且更具实际意义的可约化及不可对角化矩阵${\\bf W}$，这类矩阵具有块上三角形式，可用于建模层次化影响机制。我们首先建立同步性，证明$({\\bf Z}^\\top_n, {\\bf N}^\\top_n)^\\top \\to Z_\\infty {\\bf 1}$几乎必然成立，其中极限$Z_\\infty$的分布被证明仅由主导子群的内部动力学决定。进一步，我们建立了$({\\bf Z}_n,{\\bf N}_n)_n$的联合中心极限定理，揭示了${\\bf W}$的谱特性与若尔当块结构如何控制二阶波动。我们证明收敛速率与极限协方差结构呈现出依赖于$\\gamma$和${\\bf W}$谱特性的相变现象。关键的是，我们明确刻画了${\\bf W}$的不可对角化性如何从根本上改变渐近协方差结构，并在临界情形（$\\gamma=1$）中引入新的对数尺度因子。这些结果为基于此类层次网络结构的统计推断奠定了概率论基础。