Figure 2
Weighted-and-Reverse-Swap (WaRSwap). (A) WaRSwap weighted vertex selection step: source vertices are placed in descending degree order, and sources 'select' their targets for a directed-edge pairing of source and target. The target vertices are selected by a source via weighted random sampling without replacement, with each target being assigned an 'attraction' weight that is proportional to the current source and target capacity, but reduced by a factor that prevents overly frequent selection of large-degree source-target pairs. In this way, over-selection of hub to hub connections is avoided. In the weighting formula, m is the number of required edges in the graph. (B) WaRSwap reverse-swap step: if the algorithm reaches a state where a source S has undistributed edges to place (such as the orange edge in the top illustration) but no valid targets, S will also have unfilled capacity for at least one target (shown here as TS1) for which a valid edge exchange can be determined. The method identifies a corresponding target (shown here as TS'4) from a previously visited source S', such that the unplaced edge of S can connect to this target, and the previously placed (orange) edge from S' to TS'4 can be moved to fill the open capacity of the S' target, TS1.