HierX：百万节点网络上的快速多尺度距离衰减交互
HierX: Fast Multi-Scale Distance-Decay Interaction on Million-Node Networks
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中文总结 AI 辅助
HierX提出一种分层稀疏加校正算子，实现百万节点网络上距离衰减势场的O(n log n)快速计算，在精度和效率上优于现有方法。
中文摘要 AI 辅助
许多科学领域的模型都需要在大规模稀疏网络上进行全局距离衰减交互。引力模型、可达性度量、空间经济模型和网络影响过程都评估距离加权势场：每个位置累积来自其他所有位置的贡献，权重由它们之间最短路径旅行成本的衰减函数决定。直接计算这种聚合场需要所有成对网络成本的稠密矩阵，其时间和内存复杂度呈二次方增长。常见的近似方法要么丢弃长距离贡献，要么在交互由网络距离而非几何邻近性控制时失效。我们提出了HierX，一种用于网络上距离衰减势场的分层稀疏加校正算子。HierX构建多尺度代表性层，并带有显式校正项，确保每个位置对在最精细可用分辨率下恰好贡献一次。在空间嵌入网络中常见的有限增长假设下，应用HierX的复杂度为O(n log n)。系统性基准测试证实了在100,000个节点上的准线性扩展。在与距离截断截断和Nystrom低秩近似在25,000个区域网络上的头对头比较中，HierX以一小部分计算工作量实现了5-9%的均方根误差；Nystrom在陡峭衰减核上严重退化。案例研究计算了258万节点英国驾车网络和177万节点伦敦步行网络的人口加权可达性：一次性层次构建（约1小时）产生一个紧凑的可重用算子，然后在不到一秒内评估每个全国规模的可达性场（英国约700毫秒，伦敦约150毫秒）。提供了开源代码和实例。
英文摘要
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 multi-scale 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(n log n). Systematic benchmarks confirm quasi-linear scaling to 100,000 nodes. In head-to-head comparison with distance cutoff truncation and Nystrom low-rank approximation on 25,000-zone networks, HierX achieves 5-9% RMSE at a fraction of the computational work; Nystrom 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 (~1 hour) yields a compact reusable operator that then evaluates each national-scale accessibility field in under one second (~700 ms for Great Britain, ~150 ms for London). Open-source code and worked examples are provided.
