Our recent work [Jia et al., SIAM Rev., 57 (2015), pp. 367--397] proposes the DeGroot--Friedkin dynamical model for the analysis of social influence networks. This dynamical model describes the evolution of self-appraisals in a group of individuals forming opinions in a sequence of issues. Under a strong connectivity assumption, the model predicts the existence and semiglobal attractivity of equilibrium configurations for self-appraisals and social power in the group. In this paper, we extend the analysis of the DeGroot--Friedkin model to two general scenarios where the interpersonal influence network is not necessarily strongly connected and where the individuals form opinions with reducible relative interactions. In the first scenario, the relative interaction digraph is reducible with globally reachable nodes; in the second scenario, the condensation of the relative interaction digraph has multiple aperiodic sinks. For both scenarios, we provide the explicit mathematical formulations of the DeGroot--Friedkin dynamics, characterize their equilibrium points, and establish their asymptotic attractivity properties. This work completes the study of the DeGroot--Friedkin model with most general social network settings and predicts that, under all possible interaction topologies, the emerging social power structures are determined by the individuals' eigenvector centrality scores.