  A novel method to obtain hierarchical and overlapping clusters from network
data -i.e., a set of nodes endowed with pairwise dissimilarities- is presented.
The introduced method is hierarchical in the sense that it outputs a nested
collection of groupings of the node set depending on the resolution or degree
of similarity desired, and it is overlapping since it allows nodes to belong to
more than one group. Our construction is rooted on the facts that a
hierarchical (non-overlapping) clustering of a network can be equivalently
represented by a finite ultrametric space and that a convex combination of
ultrametrics results in a cut metric. By applying a hierarchical
(non-overlapping) clustering method to multiple dithered versions of a given
network and then convexly combining the resulting ultrametrics, we obtain a cut
metric associated to the network of interest. We then show how to extract a
hierarchical overlapping clustering structure from the aforementioned cut
metric. Furthermore, the so-called overlapping function is presented as a tool
for gaining insights about the data by identifying meaningful resolutions of
the obtained hierarchical structure. Additionally, we explore hierarchical
overlapping quasi-clustering methods that preserve the asymmetry of the data
contained in directed networks. Finally, the presented method is illustrated
via synthetic and real-world classification problems including handwritten
digit classification and authorship attribution of famous plays.
