Optimal Transport-Based Clustering of Attributed Graphs with an Application to Road Traffic Data

In many real-world contexts, such as social or transport networks, data exhibit both structural connectivity and node-level attributes. For example, roads in a transport network can be characterized not only by their connectivity but also by traffic flow or speed profiles. Understanding such systems therefore requires jointly analyzing the network structure and node attributes, a challenge addressed by attributed graph partitioning, which clusters nodes based on both connectivity and attributes. In this work, we adapt distance-based methods for this task, including Fréchet $k$-means and optimal transport-based approaches based on Gromov–Wasserstein (GW) discrepancy. We investigate how GW methods, traditionally used for general-purpose tasks such as graph matching, can be specifically adapted for node partitioning, an area that has been relatively underexplored. In the context of node-attributed graphs, we introduce an adaptation of the Fused GW method, offering theoretical guarantees and the ability to handle heterogeneous attribute types. Additionally, we propose to incorporate distance-based embeddings to enhance performance. The proposed approaches are systematically evaluated using a dedicated simulation framework and illustrated on a real-world transportation dataset. Experiments investigate the influence of target choice, assess robustness to noise, and provide practical guidance for attributed graph clustering. In the context of road networks, our results demonstrate that these methods can effectively leverage both structural and attribute information to reveal meaningful clusters, offering insights for improved network understanding.

💡 Research Summary

The paper tackles the problem of clustering nodes in graphs that carry both structural connections and rich node‑level attributes—a setting common in social, biological, and especially transportation networks. The authors focus on distance‑based approaches that can operate in arbitrary metric spaces, thereby accommodating heterogeneous attribute types such as discrete labels, histograms, and functional time‑series.

First, a unified distance between any pair of nodes is defined as a convex combination of a normalized structural distance (d_S) and an attribute distance (d_A): (d_{\alpha}= \alpha d_S + (1-\alpha)d_A). Using this distance matrix, the authors adapt the classic k‑means algorithm to graph data by replacing the Euclidean mean with a Fréchet mean, which is well‑defined for any metric space. The resulting “k‑Fréchet‑means” follows the Lloyd heuristic: assign each node to the nearest center, then recompute each center as the Fréchet minimizer of intra‑cluster distances. To mitigate sensitivity to initialization, k‑means++ seeding is employed. Computationally, the method requires pre‑computing the full (N\times N) distance matrix; assignment costs (O(Nk)) per iteration, while updating a Fréchet mean can be as expensive as (O(N^2)) in the worst case.

The core contribution lies in adapting optimal‑transport (OT) tools—specifically Gromov‑Wasserstein (GW) and its fused variant (FGW)—to the node‑partitioning task. Traditional GW measures the discrepancy between two metric measure spaces; here the authors view clustering as a transport problem from the source graph (G) to a small “target” graph (T) whose nodes represent the desired clusters. By solving a GW minimization (via entropic Sinkhorn iterations) they obtain a transport plan (\pi) that directly assigns each source node to a target node, i.e., a cluster label. Three strategies for constructing the target graph are examined: (i) a set of uniformly spaced dummy nodes, (ii) centroids obtained from a preliminary k‑means on the distance matrix, and (iii) domain‑specific partitions (e.g., known road corridors). Experiments show that data‑driven target graphs consistently yield higher Adjusted Rand Index (ARI) and Normalized Mutual Information (NMI) than naïve uniform targets.

Fused GW (FGW) further incorporates attribute information into the transport cost: the objective blends the structural GW term with an attribute term weighted by (\beta). The authors prove two propositions: (1) FGW satisfies a stronger triangle inequality than plain GW, guaranteeing that the fused distance remains a valid metric on the space of attributed graphs; (2) the alternating minimization algorithm converges to a stationary point of the FGW functional. By varying (\beta) they explore the trade‑off between topology and attributes; values around 0.5–0.7 tend to balance both sources of information and improve robustness to noisy attributes.

To alleviate the quadratic cost of GW/FGW on large graphs, the authors propose a preprocessing step that embeds each node into a low‑dimensional Euclidean space using distance‑preserving techniques (MDS, graph neural network embeddings). The embedded coordinates can be clustered with standard k‑means or supplied as the metric for GW, dramatically reducing runtime while preserving clustering quality.

The experimental section comprises two parts. On synthetic data, the authors generate non‑attributed graphs (purely structural) and attributed graphs with mixed attribute types. Results indicate that k‑Fréchet‑means performs comparably to spectral clustering on pure structure but degrades when attributes are added. GW‑based clustering with a data‑driven target outperforms all baselines, and FGW further improves ARI/NMI, especially under attribute noise (Gaussian perturbations). On a real‑world road network (the Rennes metropolitan area, ~3,000 nodes), they use traffic flow time‑series and histogram‑based speed distributions as attributes. FGW and GW with appropriate target graphs discover meaningful sub‑networks that correspond to high‑traffic corridors versus low‑traffic residential streets, confirming that the methods can simultaneously respect connectivity and traffic similarity. Visualizations show clear separation of major arteries from peripheral roads, which is not achieved by purely structural methods.

The discussion highlights several practical insights: (i) the choice of target graph is crucial; data‑driven targets adapt to the intrinsic geometry of the graph and lead to better partitions, (ii) the weighting parameters (\alpha) (for distance fusion) and (\beta) (for FGW) need modest tuning but generally work well in the mid‑range, (iii) computational cost remains a bottleneck for very large graphs; however, the proposed embedding step reduces memory and time requirements, and GPU‑accelerated Sinkhorn iterations are a promising direction. The authors also note that their framework naturally extends to multi‑modal attributes (e.g., combining categorical labels with continuous sensor readings) as long as a metric can be defined on each modality.

In conclusion, the paper delivers a comprehensive, theoretically grounded, and empirically validated suite of methods for clustering attributed graphs. By bridging Fréchet‑based k‑means with optimal‑transport formulations (GW and FGW) and augmenting them with distance‑preserving embeddings, the work offers a versatile toolkit for practitioners dealing with complex networks where both topology and node attributes matter—particularly in transportation systems where traffic dynamics must be jointly analyzed with road connectivity. Future work is suggested on streaming graphs, scalable GPU implementations, and deeper integration with graph neural networks for end‑to‑end learning of the distance metrics themselves.