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Network-wide Traffic State Estimation (TSE), which aims to infer a complete image of network traffic states with sparsely deployed sensors, plays a vital role in intelligent transportation systems. With the development of data-driven methods, traffic dynamics modeling has advanced significantly. However, TSE poses fundamental challenges for data-driven approaches, since historical patterns cannot be learned locally at sensor-free segments. Although graph representation learning shows promise in estimating states at locations without sensors, existing methods typically handle unobserved locations by filling them with zeros, introducing bias to the sensitive graph message propagation. The recently proposed Dirichlet Energy-based Feature Propagation (DEFP) method achieves State-Of-The-Art (SOTA) performance in unobserved node classification by eliminating the need for zero-filling. However, applying it to TSE faces three key challenges: inability to handle directed traffic networks, strong assumptions in traffic spatial correlation modeling, and overlooking distinct propagation rules of different patterns (e.g., congestion and free flow). We propose DGAE, a novel inductive graph representation model that addresses these challenges through theoretically derived DEFP for Directed graph (DEFP4D), enhanced spatial representation learning via DEFP4D-guided latent space encoding, and physics-guided propagation mechanisms that separately handle congested and free-flow patterns. Experiments on three traffic datasets demonstrate that DGAE outperforms existing SOTA methods and exhibits strong cross-city transferability. Furthermore, DEFP4D can serve as a standalone lightweight solution, showing superior performance under extremely sparse sensor conditions. The code of this work is publicly available at:  https://github.com/ZJU-TSELab/DGAE

Network-Wide Freeway Traffic Estimation Using Sparse Sensor Data: A Dirichlet Graph Auto-Encoder Approach