For binomial random graphs with average degree n^x, the multiset metric dimension is w.h.p. at most n^{y4} for x≤1/8 and at least n^{y1} for x≤1/2, where y1,y4 are explicit constants, and it is infinite for x>1/2.
A note on multiset dimension and local multiset dimension of graphs
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Multiset Metric Dimension of Binomial Random Graphs
For binomial random graphs with average degree n^x, the multiset metric dimension is w.h.p. at most n^{y4} for x≤1/8 and at least n^{y1} for x≤1/2, where y1,y4 are explicit constants, and it is infinite for x>1/2.