Abstract: 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.

Key words: multi-modal transportation network, traffic flow observability problem, node flow conservation, graph theory and algebra