The Idleness Parameter of Ollivier Curvature Does Not Affect Edge Selection

Preprint, two language versions (English and German). All results are fully reproducible with the accompanying scripts. Ollivier curvature on graphs depends on an idleness parameter α, the share of the random walk that stays in place. Methods for curvature-based rewiring in graph neural networks uniformly fix it at 0.5 without justifying the choice. Since the performance gains of these methods are at the same time reported to depend strongly on hyperparameters, this work tests whether the choice of α changes which edges are selected for modification. The curvature is computed with exact optimal transport at five idleness values on a grid graph, on four constructed bottleneck graphs, and on the Cora and CiteSeer benchmark datasets. The result is an all-clear: the set of the k most negatively curved edges agrees at 100 % for k ≤ 100 on both datasets. The sign of the curvature changes for 1.14 % of edges on Cora and 0.59 % on CiteSeer. The ranking across all edges shifts measurably — Kendall's τ drops to 0.769 — but the top of the ranking does not. In addition, the idleness function of the open 4×5 grid graph is given in closed form, and a methodological pitfall is reported: on small, symmetric graphs, evaluating a rule of the form "take the k smallest" produces spurious changes, because many edges share the same curvature value and sorting routines break ties arbitrarily. Deutsche Fassung. Die Ollivier-Krümmung auf Graphen hängt von einem Idleness-Parameter α ab, dem Anteil des Zufallswegs, der am Ort verbleibt. Verfahren zum curvature-based rewiring in Graph Neural Networks setzen ihn durchweg auf 0,5, ohne die Wahl zu begründen. Diese Arbeit prüft, ob die Wahl von α die Auswahl der zu ändernden Kanten verändert. Das Ergebnis ist eine Entwarnung: Die Menge der k am negativsten gekrümmten Kanten stimmt für k ≤ 100 auf beiden Benchmark-Datensätzen zu 100 % überein. Die Rangfolge über alle Kanten verschiebt sich messbar, die Spitze der Rangfolge nicht.