The ultrafast demagnetization (UFD) dynamics of itinerant ferromagnets is theoretically investigated as a function of the characteristics of the initial laser excitation. A many-body pd-band Hamiltonian is considered which takes into account hybridizations, Coulomb interactions, spin-orbit interactions, and the coupling to the laser field on the same electronic level. In this way, a fruitful connection is established between the nonadiabatic quantum dynamics and the well-known equilibrium statistical mechanics of itinerant-electron ferromagnets. The time evolution during and after the pulse absorption is determined exactly by performing numerical Lanczos propagations on a small cluster model with parameters appropriate for Ni. The most relevant laser parameters, namely, the fluence F, wave length λ, polarization ɛ, and pulse duration τp are varied systematically. The results show how F, ɛ, and τp allow one to control the total absorbed energy, the spectral distribution of the initial excitation, and the subsequent magnetization dynamics. The calculations show that reasonable changes in these parameters do not affect the UFD dynamics qualitatively and have only a minor influence on the timescale τdm which characterizes the initial demagnetization. In contrast, our model predicts that the degree of demagnetization ΔSz/S0z achieved for t≳τdm correlates well with the average number of electrons excited by the laser or average number of absorbed photons nph, which can be tuned by varying the fluence, spectral distribution and polarization of the laser pulse. The theoretical results are discussed by comparing them with available experiments. From a fundamental perspective, the robustness of the ultrafast demagnetization effect is theoretically demonstrated, as a phenomenon reflecting the intrinsic dynamics of the metallic 3d valence electrons. A wide variety of well-focused possibilities of tailoring the efficacy of the ultrafast demagnetization process is thereby opened.