The classifier chain is a widely used method for analyzing multi-labeled data sets. In this study, we introduce a generalization of the classifier chain: the classifier chain network. The classifier chain network enables joint estimation of model parameters, and allows to account for the influence of earlier label predictions on subsequent classifiers in the chain. Through simulations, we evaluate the classifier chain network's performance against multiple benchmark methods, demonstrating competitive results even in scenarios that deviate from its modeling assumptions. Furthermore, we propose a new measure for detecting conditional dependencies between labels and illustrate the classifier chain network's effectiveness using an empirical data set.