One of the main goals in the field of complex systems is the selection and extraction of relevant and meaningful
information about the properties of the underlying system from large datasets. In the last years different methods
have been proposed for filtering financial data by extracting a structure of interactions from cross-correlation
matrices where only few entries are selected by means of criteria borrowed from network theory. We discuss and compare the stability and robustness of two methods: the Minimum Spanning Tree and the Planar Maximally Filtered Graph. We construct such graphs dynamically by considering running windows of the whole dataset. We study their stability and their edges's persistence and we come to the conclusion that the Planar Maximally Filtered Graph offers a richer and more signi.cant structure with respect to the Minimum Spanning Tree, showing also a stronger stability in the long run.