Localized magnetic poles for visual analytics of multi rooted hierarchies in directed networks

Directed graphs model asymmetric relations in citation networks, software dependencies, social media, provenance, and information flow. Conventional hierarchical drawings impose one global direction, whereas standard force directed layouts emphasize clusters and centrality but commonly treat edge direction as secondary. This article presents MultiPole, a localized magnetic pole framework in which visualization users select nodes that serve as task relevant roots. Directed reachability assigns nodes and edges to pole zones. Pole gravity, optional circular pinning, localized magnetic rotation, and hop based hierarchy forces then expose several local hierarchies in one drawing. The released Cytoscape implementation places these forces in a Prefuse spring electrical simulator, while the reproducible Python benchmark uses ForceAtlas2 coordinates as a common initializer so that pole refinement can be compared under controlled conditions. We verify the mathematical formulation against the released implementation, provide a notation table and practical parameter guidance, and distinguish diagnostic metrics from forces that are actually integrated. The revised evaluation unifies the synthetic experiments into one protocol with identical graph instances, initial coordinates, seeds, iterations, and metric definitions across ablations. Across 12 paired synthetic graph instances spanning two, four, six, and eight poles, MultiPole reduces the mean bad edge fraction from 0.371 to 0.006 and the mean angular error from 52.7° to 12.1°. The same layouts increase sampled crossings from 42.3 to 115.0, confirming that local directional organization and global crossing minimization are competing objectives. The ablation shows that the structural pole and hierarchy terms account for most of the gain on this benchmark; the magnetic term alone is insufficient and contributes a smaller incremental angular improvement once stable pole geometry exists. We conclude with explicit assumption checks for citation analysis and other applications, sensitivity based parameter ranges, and limits on what the resulting geometry can support as evidence.