In this paper, we generalize the Hoede-Bakker index, which is a measure for the power of agents in a network, taking into account the mutual influences of the agents. We adopt sets of axioms different from the one adopted in the original definition. In particular, we remove an original assumption according to which changing all inclinations of the players leads to the opposite group decision. Several examples showing the usefulness of this generalization are constructed. In particular, we may apply the generalized Hoede-Bakker index to a game with a vetoer. Next, the relation between the generalized Hoede-Bakker index and the Penrose measure is analysed. Moreover, we introduce several modifications of the Hoede-Bakker index which lead to the Coleman indices, the Rae index, and the König-Bräuninger index. In order to show the relation between the generalized or the modified Hoede-Bakker index and the other power indices, we use the probabilistic approach.