In the field of network analysis, community detection plays a pivotal role. As real-world networks become increasingly complex, assuming disjoint clusters often proves too restrictive. For this reason, it becomes essential to consider more flexible models that allow for overlapping clusters. The classical overlapping stochastic block model faces limitations in explicitly controlling the degree of overlap in its results. This lack of flexibility can lead to suboptimal model fit in specific scenarios. To address this challenge, we propose an extension to the overlapping stochastic block model. Our aim is to define a model that simultaneously captures and controls for both the degree of cluster overlapping and the individual centrality of each node. Unlike the estimation procedure adopted in the classical overlapping stochastic block model, we develop a Bayesian Markov chain Monte Carlo algorithm to estimate this extended version. This work finds practical application in the analysis of the Pantheon relational network. A network derived from a biographical database of the globally famous individuals throughout human history.