Provenance Tracing in Network Diffusion Algorithms

We propose a novel strategy for provenance tracing in random walk-based network diffusion algorithms, a problem that has been surprisingly overlooked in spite of the widespread use of diffusion algorithms in biological applications. Our path-based approach enables ranking paths by the magnitude of their contribution to each node's score, offering insight into how information propagates through a network. Building on this capability, we introduce two quantitative measures: (i) path-based effective diffusion, which evaluates how well a diffusion algorithm leverages the full topology of a network, and (ii) diffusion betweenness, which quantifies a node's importance in propagating scores. We applied our framework to SARSCoV- 2 protein interactors and human PPI networks. Provenance tracing of the Regularized Laplacian and Random Walk with Restart algorithms revealed that a substantial amount of a node's score is contributed via multi-edge paths, demonstrating that diffusion algorithms exploit the non-local structure of the network. Analysis of diffusion betweenness identified proteins playing a critical role in score propagation; proteins with high diffusion betweenness are enriched with essential human genes and interactors of other viruses, supporting the biological interpretability of the metric. Finally, in a signaling network composed of causal interactions between human proteins, the top contributing paths showed strong overlap with COVID-19-related pathways. These results suggest that our path-based framework offers valuable insight into diffusion algorithms and can serve as a powerful tool for interpreting diffusion scores in a biologically meaningful context, complementing existing module- or node-centric approaches in systems biology. The code is publicly available at https://github.com/n-tasnina/provenance-tracing.git under the GNU General Public License v3.0.