网络上稀有随机动力学的条件路径蒙特卡洛方法：细节与推导
Conditional-path Monte Carlo for rare stochastic dynamics on networks: Details and derivations
AI总结：
本文提出CPMC方法，解决稀有随机网络动力学模拟的技术局限，通过严格推导与优化实现无拒绝轨迹生成，经小网络验证有效。
AI中文摘要：
对随机网络动力学中稀有宏观事件（如广泛流行的疫情暴发、通信网络中的级联故障或多体系统中从亚稳态逃逸）的模拟，受到标准前向时间算法、分裂方法和跃迁路径采样固有的灾难性拒绝率、权重简并性、谱系相关性及临界减速等方法学挑战的严重阻碍。条件路径蒙特卡洛（CPMC）通过采用直接作用于全系统轨迹的非局部Swendsen-Wang类簇更新克服了这些局限。作为[Sun, Moody和Barthel, arXiv:2608.16171]的技术配套文献，本文提供了CPMC框架的严格数学基础和算法细节：正式定义联合路径图概率权重并推导保证细致平衡的跃迁和均匀化求和规则；将框架应用于易感-感染-易感（SIS）模型，系统构建并优化专门设计以防止锁雪崩并维持疫情主干结构流动性的单节点和边图顶点集；详细说明用于精确实现复杂边界条件（包括零号患者和宏观暴发规模约束）的动态规划方案，从而实现无拒绝生成有效轨迹；最后评估算法的计算复杂度，描述并行化策略，并针对小网络上的动力学将CPMC与精确解进行验证。
英文摘要：
The simulation of rare macroscopic events in stochastic network dynamics, such as widespread epidemic outbreaks, cascading failures in communication networks, or the escape from metastable states in many-body systems, is severely hindered by methodological challenges like catastrophic rejection rates, weight degeneracy, genealogical correlations, and critical slowing down inherent to standard forward-time algorithms, splitting methods, and transition-path sampling. Conditional-path Monte Carlo (CPMC) overcomes these limitations by employing non-local Swendsen-Wang-like cluster updates that operate directly on full-system trajectories. Serving as the technical companion to [Sun, Moody, and Barthel, arXiv:2608.16171], this paper provides the rigorous mathematical foundations and algorithmic details underlying the CPMC framework. We formally define the joint path-graph probability weights and derive the transition and uniformization sum rules that guarantee detailed balance. Applying the framework to susceptible-infectious-susceptible (SIS) models, we systematically construct and optimize single-node and edge graph vertex sets specifically designed to prevent lock avalanches and maintain the structural mobility of the epidemic trunk. Furthermore, we detail a dynamic programming scheme to exactly implement complex boundary conditions - including patient-zero and macroscopic outbreak-size constraints - enabling the rejection-free generation of valid trajectories. Finally, we assess the computational complexity of the algorithm, describe parallelization strategies, and validate CPMC against exact solutions for dynamics on small networks.
