  We introduce several geometric notions, including thick-thin decompositions
and the width of a homology class, to the theory of persistent homology. These
ideas provide geometric interpretations of persistence diagrams. Indeed, we
give quantitative and geometric descriptions of the "size" or "persistence" of
a homology class. As a case study, we analyze the power filtration on
unweighted graphs, and provide explicit bounds for the life spans of homology
classes in persistence diagrams in all dimensions.
