AI 中文总结
该研究受2012年动态PageRank框架启发，引入连续时间PageRank模型，其个性化向量依记忆函数确定权重演化。通过积分 - 微分方程研究记忆函数对PageRank向量长期行为的影响，证明强连通网络中不同记忆函数下的收敛及振荡记忆导致的渐近周期行为。
AI 中文摘要
受2012年Gleich和Rossi的动态PageRank框架启发，我们引入了一个连续时间PageRank模型，其中个性化向量作为其过去值的加权平均值演化，权重由记忆函数确定。由此产生的动力学被表述为一个积分 - 微分方程的初值问题，初始条件是一个概率向量。我们研究记忆函数的选择如何影响PageRank向量的长期行为。对于强连通网络$\mathcal{G}$，我们证明了广泛的记忆函数类会导致收敛到一个与初始条件无关的稳态。相反，当记忆函数是指数振荡的，即$\omega(t)=e^{at}\cos(bt)$（$t\geq0$，$a,b>0$）时，我们表明PageRank动力学表现出渐近周期行为，揭示了振荡记忆可以从根本上改变排名过程的定性演化。为了建立这些结果，我们首先使用积分 - 微分方程理论的标准结果证明了解的存在性和唯一性，并表明解在所有时间都保持为概率向量，从而保留了PageRank模型的基本性质。
英文摘要
Inspired by the dynamical PageRank framework of Gleich and Rossi in 2012, we introduce a continuous-time PageRank model in which the personalization vector evolves as a weighted average of its past values, with the weights determined by a memory function. The resulting dynamics are formulated as an initial value problem for an integro-differential equation, where the initial condition is a probability vector. We investigate how the choice of memory function influences the long-time behavior of the PageRank vector. In particular, for strongly connected networks $\mathcal{G}$, we prove that broad classes of memory functions lead to convergence toward a stationary state that is independent of the initial condition. In contrast, when the memory function is exponential-oscillatory, $ω(t)=e^{at}\cos(bt)$ for $t\geq0$ with $a,b>0$, we show that the PageRank dynamics exhibit asymptotically periodic behavior, revealing that oscillatory memory can fundamentally alter the qualitative evolution of the ranking process. To establish these results, we first prove the existence and uniqueness of solutions using standard results from the theory of integro-differential equations and show that the solution remains a probability vector for all times, thereby preserving the essential properties of the PageRank model.
Comments31 pages, 8 figures
