  The goal of this paper is to give a diagrammatical characterization of the
information given by the Milnor invariants of links and string links. More
precisely, we describe when two string links have equal Milnor invariants of
length $\leq q$ and when a link has trivial Milnor invariants of lenght $\leq
q$. This will be done through the use of welded knot theory, involving the
notions of arrow calculus and $w_q$--concordance introduced by J-B. Meilhan and
A. Yasuhara. These results is to be compared to the classification of links up
to $C_q$--concordance obtained by J. Conant, R. Schneiderman and P. Teichner.
