Block-Preconditioned PageRank for Centrality and Intervention Analysis In Commercial Networks

Commercial networks are shaped by heterogeneous drivers (e.g., geography, category structure, and operating performance), so a single financial indicator may not reflect a node's structural role in the overall system. This protocol extracts low-dimensional feature directions using principal component analysis and applies a paired-spectrum consistency check based on a structured symplectic block matrix to improve the reliability of the selected directions. Using these directions, the workflow constructs a weighted directed graph that fuses spatial similarity, business-category synergy, and a directional-gain term derived from feature gradients. Centrality is then computed by solving the PageRank linear system with a block-preconditioned generalized minimal residual method, with explicit convergence and diagnostic checkpoints reported for reproducibility. The workflow is demonstrated on state-level retail aggregates (49 nodes) from a public dataset, where a small number of principal components captures most of the feature variance and supports stable directional weighting. Finally, the protocol evaluates a targeted strong-to-weak intervention by reallocating a quantified fraction of edge weight from high-centrality nodes to low-centrality nodes and recomputing PageRank under the same personalization setting. Overall, this protocol enables users to build an interpretable multi-source commercial network, compute numerically verified PageRank centrality, and test intervention policies with clearly defined diagnostics.