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This paper studies the geometric structure of graph-directed sets from the point of view of Lipschitz equivalence. It is proved that if ${E_i}_i$ and ${F_j}_j$ are dust-like graph-directed sets satisfying the transitivity condition, then $E_{i₁}$ and $E_{i₂}$ are Lipschitz equivalent, and $E_i$ and $F_j$ are quasi-Lipschitz equivalent when they have the same Hausdorff dimension.