AI 中文总结
研究网络 SEIR 模型，提出图诱导张量提升方法，通过从有效传播支持中选可观测量等步骤构建层次结构，该方法能降维，字典维度与网络大小线性相关，残差轨迹反映多种因素，为建模及分析控制提供基础。
AI 中文摘要
网络 SEIR 模型描述了通过接触支持的非线性传播在相互作用的亚群内部和之间的疫情传播。基于完全有序克罗内克张量的标准多项式提升产生线性高维表示，但其维度增长迅速，因为它们保留了传播图中不存在的相互作用。本文开发了一种图诱导张量提升，其可观测量从有效传播支持中选择。基于边的精确二次表示将线性隔室转换与非线性感染项分开。然后递归构建齐次层次结构。二次传播场产生下一度。线性隔室场在该度内使所得字典饱和。第一个边闭合动力学在明确的三次截断残差之前是线性的，高阶截断仅包含下一度项。第一个提升维度与亚群数量和有效传播通道成比例。在固定阶数下，在均匀有界的局部连通性下，图诱导字典随网络大小线性增长，而完全多项式提升保留与阶数相关的多项式增长。均匀的第一个边闭合残差界取决于传播率和最大加权传入传播强度。数值例证比较了每个活动通道等强度与总传入强度相等的情况。它们证实字典维度仅取决于图支持，而残差轨迹也反映了权重积累、权重分布和非线性传播。这些结果为简化建模以及随后的特定模型分析和控制提供了结构化基础。
英文摘要
Networked SEIR models describe epidemic spread within and between interacting subpopulations through contact-supported nonlinear transmission. Standard polynomial liftings based on complete ordered Kronecker tensors yield linear higher-dimensional representations, but their dimensions grow rapidly because they retain interactions absent from the transmission graph. This paper develops a graph-induced tensor lifting whose observables are selected from the effective transmission support. An exact edge-based quadratic representation separates linear compartmental transitions from nonlinear infection terms. A homogeneous hierarchy is then constructed recursively. The quadratic transmission field generates the next degree. The linear compartmental field saturates the resulting dictionary within that degree. The first edge-closure dynamics are linear up to an explicit cubic truncation residual, and higher-order truncations contain only next-degree terms. The first lifted dimension scales with the numbers of subpopulations and effective transmission channels. At fixed order, graph-induced dictionaries grow linearly with network size under uniformly bounded local connectivity, whereas complete polynomial liftings retain order-dependent polynomial growth. Uniform first edge-closure residual bounds depend on the transmission rate and the maximum weighted incoming transmission intensity. Numerical illustrations compare equal intensity per active channel with equal total incoming intensity. They confirm that dictionary dimensions depend only on graph support, whereas residual trajectories also reflect weight accumulation, weight distribution, and nonlinear propagation. These results provide a structured basis for reduced modeling and subsequent model-specific analysis and control.
Comments32 pages, 5 figures
