This study introduces a graph theory-based model that addresses the link flow observability problem in traffic networks by optimizing passive sensor deployment. The model aims to determine the minimal number of sensors and their optimal placement. It constructs a virtual network and uses isomorphic graph theory to map between the original and virtual networks, ensuring consistency in nodes, links, and link directions. Two formulas are proposed to calculate the minimum number of observable links required across different networks, factoring in links, ordinary nodes, centroid nodes, and added links. Key concepts such as chords, cut sets, and loops, along with their matrices, are analyzed. A matrix-based framework is developed to consider flow conservation conditions. Results show that solving the full link flow observability problem using node flow conservation equations yields a fixed number of sensors with non-unique deployment schemes, Additionally, a resource-constrained sensor network optimization (RSNO) model is presented, employing null space projection (NSP) as an objective function to quantify the impact of budget constraints particularly under the condition if all the link flows cannot be observed. Numerical examples demonstrate the RSNO model's applications.

中文翻译：

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重新审视流量可观测性问题：基于矩阵的模型，适用于具有或不具有质心节点的流量网络


本研究介绍了一种基于图论的模型，该模型通过优化无源传感器部署来解决流量网络中的链接流可观测性问题。该模型旨在确定传感器的最小数量及其最佳放置位置。它构建了一个虚拟网络，并使用同构图论在原始网络和虚拟网络之间进行映射，确保节点、链接和链接方向的一致性。提出了两个公式来计算不同网络中所需的最小可观察链接数，包括链接、普通节点、质心节点和添加的链接。分析了和弦、剪辑集和 Loop 等关键概念及其矩阵。开发了一个基于矩阵的框架来考虑流动守恒条件。结果表明，使用节点流守恒方程求解全链路流可观测性问题，可以产生固定数量的具有非唯一部署方案的传感器，此外，还提出了一个资源约束传感器网络优化 （RSNO） 模型，采用零空间投影 （NSP） 作为目标函数来量化预算约束的影响，特别是在无法观察到所有链路流的情况下。数值示例演示了 RSNO 模型的应用。