HSTE-GNN for City-Scale Dynamic Routing

• The paper introduces a scalable distributed framework that uses graph partitioning (via METIS) to efficiently manage city-scale dynamic road networks.
• It features an edge-enhanced spatio-temporal module that employs dynamic edge updates and message passing to capture localized traffic fluctuations and long-term congestion patterns.
• The model’s hierarchical synchronization aggregates regional summaries to ensure global routing consistency, achieving near-linear scalability with minimal accuracy loss.

A Distributed Hierarchical Spatio-Temporal Edge-Enhanced Graph Neural Network (HSTE-GNN) is a scalable deep learning architecture designed to address city-scale dynamic logistics routing problems over ultra-large, fast-evolving urban road networks. HSTE-GNN is characterized by three major components: distributed graph partitioning and parallelization, an edge-enhanced spatio-temporal graph neural module, and a hierarchical synchronization protocol that ensures global coherence under real-time traffic conditions. It enables efficient learning of both localized traffic dynamics and long-range congestion patterns, executing inference and training on graphs with millions of nodes and edges, under dynamic traffic updates and large-scale logistics workloads (Han et al., 20 Dec 2025).

1. Distributed Architecture and Graph Partitioning

At each time step $t$, the city-scale road network is formulated as a dynamic graph $G_t = (V, E_t)$, where $V$ includes all intersections, logistic depots, delivery/pick-up points, and vehicle positions, and $E_t$ encompasses all road segments with time-varying edge attributes $e_{ij}^t$ reflecting speeds, flows, and incidents.

To ensure scalability and efficient resource utilization, HSTE-GNN employs graph partitioning via METIS, dividing $G_t$ into $R$ disjoint geographic subgraphs:

$$G_t^{(r)} = (V^{(r)}, E_t^{(r)}), \quad r = 1, \dots, R$$

Each region is allocated to a dedicated compute node (e.g., GPU server), maintaining local node features ($x_i^t$) and edge features ($e_{ij}^t$ for $(i,j)\in E_t^{(r)}$). This method dramatically reduces per-node memory requirements and leverages parallel computation for both training and inference.

2. Edge-Enhanced Spatio-Temporal Module

Within each region, an edge-enhanced spatio-temporal GNN (EE-STGNN) module jointly models:

• Node states: $h_i^t$
• Time-varying edge attributes: $e_{ij}^t$
• Short-term temporal dependencies

The update procedure at every time step $t$ consists of:

1. Dynamic Edge Update:

$$e_{ij}^t = \Psi_e\left(e_{ij}^{t-1}, h_i^{t-1}, h_j^{t-1}, x_{ij}^t\right)$$

Here, $x_{ij}^t$ are the newest traffic measurements, and $\Psi_e$ is an MLP that captures minute-level travel-time fluctuations.

2. Edge-Aware Message Passing:

$$m_{ij}^t = \Phi_m\left(h_i^{t-1}, h_j^{t-1}, e_{ij}^{t-1}\right)$$

$\Phi_m$ fuses node and edge histories to capture the influence of both neighboring nodes and current traffic.

3. Node State Update:

$$h_i^t = \Phi_u\left(h_i^{t-1}, \sum_{j \in N(i)} m_{ij}^t\right)$$

$\Phi_u$ (typically an MLP or GRU-type update) incorporates self-history and aggregated edge-aware messages to compute the new node embedding.

This edge-centric message passing enables the model to directly track the evolution of critical traffic characteristics at the road segment level, distinguishing HSTE-GNN from node-only approaches.

3. Hierarchical Aggregation and Global Synchronization

After a fixed number of local EE-STGNN layers ($K$ steps, e.g., $K=5$), each region computes a region summary $S^{(r)}$ using attention-based pooling:

$$\beta_i = \frac{\exp(u^\top\tanh(W h_i^t))}{\sum_{v\in V^{(r)}}\exp(u^\top\tanh(W h_v^t))}$$

$S^{(r)} = \sum_{i\in V^{(r)}} \beta_i h_i^t$

All regional summaries $\{ S^{(1)}, \ldots, S^{(R)} \}$ are then aggregated asynchronously via a parameter server (PS) or an AllReduce protocol to obtain a global context vector:

$$g^t = \mathrm{AttentionPool}_\text{global}\left(\{ S^{(1)}, \dots, S^{(R)} \}\right)$$

This asynchronous “push-pull” synchronization mechanism offers a crucial balance: immediate regional adaptation to fresh local traffic data, and periodic injection of global congestion/topology information to all regions, thus maintaining consistent city-wide routing even as the system experiences high-frequency updates.

4. Distributed Training and Inference Pipeline

Each compute node processes its assigned region’s subgraph, independently running EE-STGNN layers on new traffic feeds arriving every 5–60 seconds. After every $K$ local updates, region summaries are sent to the parameter server:

• Training involves asynchronous AllReduce every 5 (forward–backward) steps, with gradients or region embeddings exchanged only at the summary granularity ($O(Rd)$), not at the full-graph level ($O(|E|)$), thus reducing bandwidth and central processing requirements.
• Inference similarly leverages this low-overhead synchronization for online city-scale deployment.

This distributed design allows HSTE-GNN to preserve low-latency reaction to localized events, while enforcing city-wide routing quality. Empirically, this pipeline yields near-linear scaling proportional to the number of available GPU nodes, with accuracy losses below 1% at 32-fold parallelization.

5. Experimental Setup

Experiments were conducted on the following real-world datasets and cluster configuration:

Dataset/Cluster Property	Value
Beijing Road Network	$\sim$1.2M nodes, 2.4M edges, 6 months of 5-min traffic reading and courier traces
New York City Network	$\sim$0.8M nodes, 1.6M edges, similar traffic & delivery logs
Compute Cluster	16 GPU nodes (NVIDIA A100, 256GB), 32 CPU nodes, 100 Gbps InfiniBand
Partitioning/Assignment	METIS, $R=32$ regions, 1 region per GPU node
Batch Size	64 temporal sequences/region
Optimizer	AdamW, 100 epochs
Synchronization Module	Asynchronous AllReduce every 5 local steps
Test Split	Final 20% of last 30 days’ data

Baseline models included GCN, GAT, T-GCN, DCRNN, and ST-GRAPH, with evaluation using both travel-time prediction metrics (RMSE, MAE, MAPE, $R^2$) and routing metrics (OPD: Optimal Path Deviation in minutes, RCS: Route Consistency Score).

6. Quantitative Results and Ablation Analysis

On both Beijing and New York datasets, HSTE-GNN achieved substantial improvements over the best spatio-temporal baseline (ST-GRAPH):

Metric	HSTE-GNN	ST-GRAPH	Relative Improvement
RMSE	5.48	6.21	–11.8%
MAE	4.12	4.86
MAPE	8.7%	10.2%	–14.7%
$R^2$	0.884	0.846
OPD (min)	2.31	3.55	–34.9% (routing delay)
RCS	0.851	0.793	+7.3% (route consistency)

Ablation studies revealed:

• Removing edge updates ($\Psi_e$) increased RMSE to 6.02 and OPD by 18%.
• Disabling the hierarchical (global) synchronization reduced RCS by 4%.
• Computation scaled nearly linearly: a 32$\times$ speedup on 32 GPUs cost less than 1% accuracy loss.

7. Significance and Implications

The HSTE-GNN framework demonstrates that distributing spatio-temporal GNNs over regional partitions, while maintaining global consensus via asynchronous synchronization, is effective for modeling real-time, city-scale dynamic logistics. The edge-enhanced message passing and temporal modeling enable rapid adaptation to sub-minute traffic perturbations, while the hierarchical aggregation layer guarantees overall routing consistency and accuracy. These results indicate that HSTE-GNN is a viable solution for next-generation intelligent transportation systems and large logistics platforms, especially as urban road networks continue to scale (Han et al., 20 Dec 2025).