Aqueous solutions of ionic liquids are of special interest, due to the distinctive properties of ionic liquids, in particular, their amphiphilic character. A better understanding of the structure-property relationships of such systems is hence desirable. One of the crucial molecular-level interactions that influences the macroscopic behavior is hydrogen bonding. In this work, we conduct molecular dynamics simulations to investigate the effects of ionic liquids on the hydrogen-bond network of water in dilute aqueous solutions of ionic liquids with various combinations of cations and anions. Calculations are performed for imidazolium-based cations with alkyl chains of different lengths and for a variety of anions, namely, [Br](-), [NO3](-), [SCN](-) [BF4](-), [PF6](-), and [Tf2N](-). The structure of water and the water-ionic liquid interactions involved in the formation of a heterogeneous network are analyzed by using radial distribution functions and hydrogen-bond statistics. To this end, we employ the geometric criterion of the hydrogen-bond definition and it is shown that the structure of water is sensitive to the amount of ionic liquid and to the anion type. In particular, [SCN](-) and [Tf2N](-) were found to be the most hydrophilic and hydrophobic anions, respectively. Conversely, the cation chain length did not influence the results.