{"ID":2845432,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2511.04562","arxiv_id":"2511.04562","title":"Asymptotics for Reinforced Stochastic Processes on Hierarchical Networks","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.","short_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...","url_abs":"https://arxiv.org/abs/2511.04562","url_pdf":"https://arxiv.org/pdf/2511.04562v2","authors":"[\"Li Yang\",\"Dandan Jiang\",\"Jiang Hu\",\"Zhidong Bai\"]","published":"2025-11-06T17:17:36Z","proceeding":"math.ST","tasks":"[\"math.ST\"]","methods":"[]","has_code":false}