HierX: Fast multiscale distance-decay interaction on million-node networks

Many models across the sciences require global distance-decay interactions on large sparse networks. Gravity models, accessibility measures, spatial economic models, and network influence processes all evaluate distance-weighted potential fields: each location accumulates contributions from every other location, weighted by a decaying function of the shortest-path travel cost between them. Computed directly, such aggregate fields require the dense matrix of all pairwise network costs, which scales quadratically in time and memory. Common approximations either discard long-range contributions or fail when interaction is governed by network distances rather than geometric proximity. We introduce HierX, a hierarchical sparse-plus-correction operator for distance-decay potential fields on networks. HierX constructs multiscale representative layers with explicit correction terms ensuring each location pair contributes exactly once at the finest available resolution. Under bounded-growth assumptions common in spatially embedded networks, applying HierX scales as $O(nlogn)$ . Systematic benchmarks confirm quasi-linear scaling to 100,000 nodes. In head-to-head comparison with distance cutoff truncation and Nyström low-rank approximation on 25,000-zone networks, HierX achieves 5-9% root-mean-square error (RMSE) at a fraction of the computational work; Nyström degrades severely on steep decay kernels. Case studies compute population-weighted accessibility on the 2.58-million-node Great Britain driving network and the 1.77-million-node London pedestrian network: a one-time hierarchy construction ( $\sim 1$ h) yields a compact reusable operator that then evaluates each national-scale accessibility field in under 1 s ( $\sim 700$ ms for Great Britain, $\sim 150$ ms for London). Open-source code and worked examples are provided.