Link Prediction for Flow-Driven Spatial Networks

TL;DR

This work tackles link prediction in flow-driven spatial networks by introducing Graph Attentive Vectors (GAV), a framework that uses $h$-hop subgraph extraction, line-graph vector embeddings, and an attentive, constrained update rule to imitate simplified physical flow. The method combines a labeling trick and a dedicated readout to produce interpretable, physically plausible predictions, optimized with binary cross-entropy loss. Empirically, GAV achieves state-of-the-art performance across eight datasets (notably surpassing ogbl-vessel baselines with 98.38 AUC vs 87.98 and far fewer parameters) and demonstrates robust ablations that validate its components and the importance of the flow-inspired inductive bias. The results have practical implications for refining vascular and road-network graphs and suggest future directions to integrate conservation laws and invariances for deeper physical fidelity.

Abstract

Link prediction algorithms aim to infer the existence of connections (or links) between nodes in network-structured data and are typically applied to refine the connectivity among nodes. In this work, we focus on link prediction for flow-driven spatial networks, which are embedded in a Euclidean space and relate to physical exchange and transportation processes (e.g., blood flow in vessels or traffic flow in road networks). To this end, we propose the Graph Attentive Vectors (GAV) link prediction framework. GAV models simplified dynamics of physical flow in spatial networks via an attentive, neighborhood-aware message-passing paradigm, updating vector embeddings in a constrained manner. We evaluate GAV on eight flow-driven spatial networks given by whole-brain vessel graphs and road networks. GAV demonstrates superior performances across all datasets and metrics and outperformed the state-of-the-art on the ogbl-vessel benchmark at the time of submission by 12% (98.38 vs. 87.98 AUC). All code is publicly available on GitHub.

Figures (10)

• Figure 1: Flow-driven spatial network $\mathcal{G}$ (right), representing vasculature (left). $\mathcal{G}$'s nodes are embedded in a Euclidean space and represent spatial positions specified by $x$-, $y$-, and $z$-coordinates.
• Figure 2: Overview of the GAV link prediction framework. GAV is divided into three modules, namely the subgraph extraction module, the message-passing module, and the readout module. First, an $h$-hop enclosing subgraph $\mathcal{G}^{t}_{h}$ is extracted around the target nodes $\{ n_{i}^{t}, n_{j}^{t} \}$ (red and green) affiliated to the target link $e^{t}_{ij}$ (orange) and subsequently transformed into a line graph representation $\mathcal{L}(\mathcal{G}^{t}_{h})$ to construct vector embeddings representative of the network's structural properties; second, we perform iterative message-passing among vector embeddings in $\mathcal{L}(\mathcal{G}^{t}_{h})$ via $k$ GAV layers, modeling simplified physical flow in spatial networks; and, third, a final subgraph-level readout module aggregates refined vector embeddings and predicts the probability of link existence with regard to the target link $e^{t}_{ij}$. To provide a concise visualization, $h$ was set to 1. We would like to draw the reader's attention to color coding.
• Figure 3: Visualization of a single GAV layer updating the vector embedding of node $n'_{j}$. We forward the vector embedding of node $n'_{j}$ together with $N_{j} \in \mathbb{R}^{| \mathcal{N}(n'_j) \cup {n'_j}| \times d_{\text{spatial}}}$, which represents the set of $n'_{j}$ and its neighbors $n'_{k}$ and $n'_{i}$, to the GAV layer. In this specific example, the vector embedding has been flipped (see $\hat{n}'_{j}$), which can be interpreted as a change in direction of physical flow.
• Figure 4: Labels generated by the labeling trick for an exemplary line graph ($h$ set to two). Our interpretation of the labeling trick assigns the label 0 to the vector embedding representing the target link (orange), the labels 1 and 2 to vector embeddings representing edges connected to the target nodes $n^t_{i}$ and $n^t_{j}$ (purple and blue), and the label 3 to remaining vector embeddings.
• Figure 5: Visualization of the effect of our GAV layer on vector embeddings (ogbl-vessel). We visualize subgraph representations $\mathcal{G}^t_h$ ($h$ set to one) of four positive, plausible (first row) and four negative, implausible target links (second row) together with the GAV layer's predicted scalar values $s_{i} \in (-1, 1)$. The scalar values $s_{i}$ used to update vector embeddings in $\mathcal{L}(\mathcal{G}^t_h)$ have been projected to $\mathcal{G}^t_h$ to provide an interpretable visualization. The directionality of edges already incorporates potential shifts in direction enforced by our GAV layer. Please note that following the color coding scheme of Fig. \ref{['fig:method_overview']}, the target link $e^{t}_{ij}$ is depicted in orange, whereas the two target nodes $n_{i}^{t}$ and $n_{j}^{t}$ are displayed in red and green. We additionally report the angle $\angle$ between vector embeddings aggregated around the two target nodes (see Section \ref{['ref:readout']}) and the predicted probability of link existence $\hat{y}^t_{ij}$. The last column contains more challenging cases.