In a hyperbolic embedding, radius is meant to say how central a node is. A model that learns radius freely learns it in one graph's scale, so the same structural role lands at different radii in different graphs. Invariant Hyperbolic Unfolding fixes radius from a structural percentile inside each graph and lets learning move only the angle. Generate two graphs of different size and see what that changes.
Radius is supposed to encode hierarchy, but nothing calibrates it across disjoint graphs. Switch to "raw score": a radius set on graph A's scale puts graph B's hubs and periphery at different radii than A's, and a frozen distance-based decoder has no way to tell which scale it is looking at.
Each node's coreness (or degree, PageRank) becomes a mid-rank percentile inside its own graph, with ties sharing a rank. One monotone map, r = Rmax(1 − p)β, turns the percentile into a radius. Equal percentile means equal radius in every graph.
Message passing averages tangent vectors, and averaging vectors that point different ways shortens them. Turn Continuation off and add layers: the radial channel drifts and flattens. With it on, each layer keeps the new direction and resets the radius to its shell.