Let H, Kbe Hilbert spaces over Fwith dimH≥3, where Fis the real or complex field. Assume that φ:B(H)→B(K)is an additive surjective map and r≥3is a positive integer. It is shown that φis r-nilpotent perturbation of scalars preserving in both directions if and only if either φ(A)=cTAT-1+g(A)Iholds for every A∈B(H); or φ(A)=cTA∗T-1+g(A)Iholds for every A∈B(H), where 0≠c∈F, T:H→Kis a τ-linear bijective map with τ:F→Fan automorphism and gis an additive map from B(H)into F. As applications, for any integer k≥5, additive k-commutativity preserving maps and general completely k-commutativity preserving maps on B(H)are characterized, respectively.