标题： 网络上的瞬态随机行走者：到达时间、预算更新和生存动力学 显示英文标题

标题： Evanescent random walker on networks: Hitting times, budget renewal, and survival dynamics

摘要： 我们考虑一个在网络上以离散时间演化的死亡随机行走者，其中转移遵循一种度数偏好的马尔可夫导航策略。 行走者从一个随机的初始预算$T_1 \in \mathbb{N}$开始，并必须保持严格正的预算以继续存活。 每一步都会产生一个单位成本，使预算减少一；当预算耗尽时，行走者会死亡（被摧毁）。 然而，当行走者到达指定的目标节点时，预算会通过一个独立同分布（IID）的初始值副本进行重置。 度数偏差被调整为偏向或不偏向访问这些目标节点。 我们的模型与随机重置存在联系。 预算的演化可以解释为在整数线上向负值方向的确定性漂移，其中行走者间歇性地被重置到正的 IID 随机位置，并在第一次到达原点时死亡。 论文的第一部分关注永生马尔可夫行走者的靶点击中统计特性。 我们分析了任意一组目标节点的\textit{目标击中计数过程}（THCP）。 在此框架下，论文的第二部分研究了易逝行走者的动态特性。 我们推导了目标节点任意配置下的解析结果，包括易逝传播矩阵、生存概率、行走者生命周期内在一个节点集合上的平均停留时间以及预期寿命本身。 此外，我们计算了行走者一生中预期的目标击中次数（即预算重置次数）及相关分布。 我们通过分析和数值方法探讨了影响行走者预期寿命的各种场景。 显示英文摘要

摘要： We consider a mortal random walker evolving with discrete time on a network, where transitions follow a degree-biased Markovian navigation strategy. The walker starts with a random initial budget $T_1 \in \mathbb{N}$ and must maintain a strictly positive budget to remain alive. Each step incurs a unit cost, decrementing the budget by one; the walker perishes (is ruined) upon depletion of the budget. However, when the walker reaches designated target nodes, the budget is renewed by an independent and identically distributed (IID) copy of its initial value. The degree bias is tuned to either favor or disfavor visits to these target nodes. Our model exhibits connections with stochastic resetting. The evolution of the budget can be interpreted as a deterministic drift on the integer line toward negative values, where the walker is intermittently reset to positive IID random positions and dies at the first hit of the origin. The first part of the paper focuses on the target-hitting statistics of an immortal Markovian walker. We analyze the \textit{target hitting counting process} (THCP) for an arbitrary set of target nodes. Within this framework, the second part of the paper addresses the dynamics of the evanescent walker. We derive analytical results for arbitrary configurations of target nodes, including the evanescent propagator matrix, the survival probability, the mean residence time on a set of nodes during the walker's lifetime, and the expected lifetime itself. Additionally, we compute the expected number of target hits (i.e., budget renewals) in a lifetime of the walker and related distributions. We explore both analytically and numerically various scenarios affecting the life expectancy of the walker.