基于图论与代数系统理论,本文针对多模式交通网络的可观测性问题建立统一的数学分析框架。多模式交通系统中交通方式异质性叠加与换乘行为复杂耦合带来结构复杂性,本文融合多层网络与超图理论,提出同构多层虚拟网络建模方法。通过引入虚拟节点与虚拟路段,构建集“结构可描述性”与“状态可观测性”于一体的统一拓扑代数理论,将公交、小汽车及换乘弧整合在同一拓扑结构中,并将含质心节点的复杂网络映射为满足流量守恒的虚拟网络,为后续分析提供统一、可计算的代数基础。在此框架基础上,本文系统推导节点法与面法的可观测性判定定理,建立最小观测数与节点数、边数、面数等拓扑参数间的显式数学关系,将“最少需要布设多少传感器”这一复杂问题转化为简洁的代数表达式。进一步提出虚拟网络构建与传感器优化布设算法,完整回答“最少需要多少传感器”以及“它们应布设于何处”这两个核心问题。该建模与求解过程本质上构成一类带拓扑约束的组合优化问题,体现图论结构与运筹学决策范式的深度融合。数值算例表明,该理论框架为多模式交通网络提供从拓扑描述到状态感知的完整解决方案,实现最小传感器集合的精确求解与系统状态的完全可观测,为提升多模式交通系统运行效率提供理论基础与实践路径。
Based on graph theory and algebraic system theory, this paper establishes a unified mathematical analysis framework for the observability problem of multimodal transportation networks. The structural complexity arising from the heterogeneity of transportation modes and the complex coupling of transfer behaviors is addressed by integrating multilayer network and hypergraph theories, which leads to the proposal of an isomorphic multilayer virtual network modeling method. Through the introduction of virtual nodes and virtual links, a unified topological-algebraic theory integrating both "structural describability" and "state observability" is constructed. This approach consolidates buses, cars, and transfer arcs within a unified topological structure and maps complex networks containing centroid nodes into virtual networks that satisfy flow conservation, thereby providing a unified and computable algebraic foundation for subsequent analysis. Building upon this framework, this paper systematically derives observability determination theorems based on the node and face methods, establishing explicit mathematical relationships between the minimum number of required observations and topological parameters such as the number of nodes, links, and planes. This transforms the complex problem of "determining the minimum number of sensors to deploy" into concise algebraic expressions. Furthermore, a virtual network construction and sensor optimal deployment algorithm is proposed, providing complete answers to the two core questions: "what is the minimum number of sensors required" and "where should they be deployed." This modeling and solution process essentially constitutes a class of combinatorial optimization problems with topological constraints, reflecting the deep integration of graph-theoretic structures and operational research decision paradigms. Numerical examples demonstrate that the proposed theoretical framework provides a complete solution from topological description to state perception for multimodal transportation networks, achieving precise identification of the minimum sensor set and complete observability of the system state. This work provides a theoretical foundation and a practical pathway for enhancing the operational efficiency of multimodal transportation systems.
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