MODELING ADVECTION ON DISTANCE-WEIGHTED DIRECTED NETWORKS

The paper’s existence–uniqueness characterization and the resulting coefficients a_uv agree with the model’s derivation and are correct, using Advection II to fix ∑_u d_vu a_uv = −1 and Splitting to enforce a_uv ∝ 1/d_vu, yielding a_uv = −1/(deg^+(v) d_vu) off-diagonal and the corresponding diagonal a_vv (Theorem 1; proof sketch aligns with Propositions 4, 5 and Corollary 2) . However, the paper’s claimed ℓ∞ bound “∥A_G f∥_∞ = 2 sup_u |a_uu| ∥f∥_∞ ≤ 2/δ_G ∥f∥_∞” (eq. (15)) is incorrect: 2 sup |a_uu| is the column-sum (ℓ1→ℓ1) bound, not the row-sum (ℓ∞→ℓ∞) bound, and the follow-on Remark 6 (extending to unbounded degrees) therefore does not hold as stated . The model correctly identifies the ℓ1 bound ∥A_G∥_{1→1} ≤ 2/δ_G and the ℓ∞ bound ∥A_G∥_{∞→∞} ≤ (1+Δ_G)/δ_G (using the degree bound from Definition 6) and provides a counterexample (an in-star) showing that a graph-uniform 2/δ_G ℓ∞ bound fails without an in-degree bound .