Formula

Hop-Penalized Path Score in Bidirectional Diffusion

In the bidirectional-diffusion retriever, for a path pp from a dense seed uu to a target node vv, the upward and downward path scores are

s(p)=s0(u)δp,s(p)=s0(u)δchildp,s_\uparrow(p) = s_0(u) - \delta\,|p|,\qquad s_\downarrow(p) = s_0(u) - \delta_{\text{child}}\,|p|,

where s0(u)s_0(u) is the seed's dense score, p|p| is the path length in edges, and δ\delta (upward) and δchild\delta_{\text{child}} (downward) are role-specific additive hop penalties. The scores are linear in path length, so each additional hop subtracts a fixed amount from the seed's dense score.

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Updated 2026-05-18

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