Expanding the Katz Index for Link Prediction: A Case Study on a Live Fish Movement Network

TL;DR

The study addresses the limitation of static link prediction in aquaculture disease modelling by extending the Katz Index to incorporate spatial distance and temporal dynamics. It introduces the Weighted Katz Index (WKI) and Edge Weighted Katz Index (EWKI), along with hybrid variants, and formulates the corresponding mathematical expressions, e.g., $KI_{(u,v)} = \sum_{l=1}^{\infty} \beta^l (A^l)_{(u,v)}$ and $\omega_{(u,v)} = e^{-\gamma \times d_{(u,v)}}$, to capture distance effects. EWKI achieves superior predictive performance with precision 0.988, recall 0.712, F1 0.827, and AUPR 0.970, while AUROC reaches 1.000 for EWKI and WKIEWKI, underscoring the value of spatial-temporal integration for disease spread prediction. The results demonstrate that geographic proximity and temporal movement patterns substantially improve link prediction in live-fish networks, offering actionable insights for surveillance and biosecurity planning. Future work will explore richer farm features, graph neural networks, and network rewiring under node removals to further enhance robustness and adaptability.

Abstract

In aquaculture, disease spread models often neglect the dynamic interactions between farms, hindering accuracy. This study enhances the Katz index (KI) to incorporate spatial and temporal patterns of fish movement, improving the prediction of farms susceptible to disease via live fish transfers. We modified the Katz index to create models like the Weighted Katz Index (WKI), Edge Weighted Katz Index (EWKI), and combined models (e.g., KIEWKI). These incorporate spatial distances and temporal movement patterns for a comprehensive aquaculture network connection prediction framework. Model performance was evaluated using precision, recall, F1-scores, AUPR, and AUROC. The EWKI model significantly outperformed the traditional KI and other variations. It achieved high precision (0.988), recall (0.712), F1-score (0.827), and AUPR (0.970). Combined models (KIEWKI, WKIEWKI) approached, but couldn't surpass, EWKI performance. This study highlights the value of extending Katz index models to improve disease spread predictions in aquaculture networks. The EWKI model's performance demonstrates an innovative and flexible approach to tackling spatial challenges within network analysis.

Figures (2)

• Figure 1: Temporal network graph of live fish movements over time. The nodes represent individual fish farms. Solid lines indicate observed movements of fish between farms within the time interval ($t_i$ to $t_j$) and dotted lines denote the predicted future movements for the interval. The horizontal axis represents the timestamps, showcasing the evolution of the network: training ($G_{train}$), validation ($G_{val}$), and test ($G_{test}$) set.
• Figure 2: (a) ROC curves depicting the performance of Katz index models with their respective AUC values, showcasing their ability to distinguish between true and false positives. (b) PR curves representing the precision-recall trade-offs for the same models, highlighting their performance in the context of class imbalance, as measured by AUPR values.