In this paper we investigate clustering in the weighted setting, in which
every data point is assigned a real valued weight. We conduct a theoretical
analysis on the influence of weighted data on standard clustering algorithms in
each of the partitional and hierarchical settings, characterising the precise
conditions under which such algorithms react to weights, and classifying
clustering methods into three broad categories: weight-responsive,
weight-considering, and weight-robust. Our analysis raises several interesting
questions and can be directly mapped to the classical unweighted setting.