AI 中文总结
研究一族演化网络模型的概率生成函数方程，通过对超临界高尔顿 - 沃森过程自包含局部分析，确定其极限度分布系数尾，解决了Feng等人的两个猜想，包括幂律阶及乘性周期形式。
AI 中文摘要
对于固定整数\(j\geq1\)和\(0\lt p\lt1\)，研究概率生成函数（pgf）方程\((1 + 2p)g(x)=2px^{j}+g(x - px + px^{2})\)，\(0\leq x\leq1\)，它控制一族演化网络模型的极限度分布\(\{p_k\}\)。\(j = 1\)和\(j = 2\)的情况分别是Feng和Hu的树状快速增长模型以及Feng、Li和Hu的均匀演化网络。证明了对于每个\(j\)，方程有唯一pgf解，均值为\(2j\)，并精确确定其系数尾。解决了Feng及其合作者的两个猜想：幂律阶\(p_k=\Theta(k^{-1 - \rho})\)及其细化到乘性周期形式\(p_k\sim\Psi_j(\log_{\lambda}k)k^{-1 - \rho}\)。方法是对具有后代律\(1+\mathrm{Bernoulli}(p)\)的超临界高尔顿 - 沃森过程进行自包含局部分析，并在独立几何时间进行检查。主要结果由多智能体系统Eureka获得并经作者验证。
英文摘要
For a fixed integer $j\ge1$ and $0<p<1$, we study the probability generating function (pgf) equation \[ (1+2p)\,g(x)=2p\,x^{j}+g\bigl(x-px+px^{2}\bigr),\qquad 0\le x\le1 , \] which governs the limiting degree distribution $\{p_k\}$ of a family of evolving network models. The cases $j=1$ and $j=2$ are the treelike fast-growth model of Feng and Hu and the homogeneous evolving network of Feng, Li and Hu. We prove that for every $j$ the equation has a unique pgf solution, of mean $2j$, and we determine its coefficient tail exactly: \[ p_k=k^{-1-ρ}\,Ψ_j(\log_λk)+o\bigl(k^{-1-ρ}\bigr), \] where $λ=1+p$, $ρ=\log(1+2p)/\log(1+p)$ is independent of $j$, and $Ψ_j$ is continuous, strictly positive and $1$-periodic, with explicit Fourier coefficients. This resolves two conjectures of Feng and coauthors: (1) the power-law order $p_k=Θ(k^{-1-ρ})$ and (2) its refinement to the multiplicatively periodic form $p_k\simΨ_j(\log_λk)\,k^{-1-ρ}$. The periodic factor is genuinely non-constant for $p$ near $1$, and, for the two network models, for all $p$ outside a discrete set. Consequently, $p_k$ is asymptotic to no constant multiple of $k^{-1-ρ}$. Our method is a self-contained local analysis of the supercritical Galton-Watson process with offspring law $1+\mathrm{Bernoulli}(p)$, inspected at an independent geometric time. This time-changed process solves the equation observed by Feng and coauthors. The main results of this paper were obtained by the multi-agent system Eureka and have subsequently been verified by the authors.
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