We consider a coarse-graining of high-dimensional potential energy landscapes basedupon persistences, which correspond to lowest barrier heights to lower-energy min-ima. Persistences can be calculated efficiently for local minima in kinetic transitionnetworks that are based on stationary points of the prevailing energy landscape. Thenetworks studied here represent peptides, proteins, nucleic acids, an atomic cluster,and a glassy system. Minima with high persistence values are likely to representsome form of alternative structural morphology, which, if appreciably populated atthe prevailing temperature, could compete with the global minimum (defined as in-finitely persistent). Threshold values on persistences (and in some cases equilibriumoccupation probabilities) have therefore been used in this work to select subsets ofminima, which were then analysed to see how well they can represent features ofthe full network. Simplified disconnectivity graphs showing only the selected minimacan convey the funnelling (including any multiple-funnel) characteristics of the corre-sponding full graphs. The effect of the choice of persistence threshold on the reduceddisconnectivity graphs was considered for a system with a hierarchical, glassy land-scape. Sets of persistent minima were also found to be useful in comparing networksfor the same system sampled under different conditions, using minimum orientedspanning forests.