Centrality, which quantifies the "importance" of individual nodes, is among
the most essential concepts in modern network theory. Most prominent centrality
measures can be expressed as an aggregation of influence flows between pairs of
nodes. As there are many ways in which influence can be defined, many different
centrality measures are in use. Parametrized centralities allow further
flexibility and utility by tuning the centrality calculation to the regime most
appropriate for a given network. Here, we identify two categories of centrality
parameters. Reach parameters control the attenuation of influence flows between
distant nodes. Grasp parameters control the centrality's potential to send
influence flows along multiple, often nongeodesic paths. Combining these
categories with Borgatti's centrality types [S. P. Borgatti, Social Networks
27, 55-71 (2005)], we arrive at a novel classification system for parametrized
centralities. Using this classification, we identify the notable absence of any
centrality measures that are radial, reach parametrized, and based on acyclic,
conservative flows of influence. We therefore introduce the ground-current
centrality, which is a measure of precisely this type. Because of its unique
position in the taxonomy, the ground-current centrality has significant
advantages over similar centralities. We demonstrate that, compared to other
conserved-flow centralities, it has a simpler mathematical description.
Compared to other reach centralities, it robustly preserves an intuitive rank
ordering across a wide range of network architectures. We also show that it
produces a consistent distribution of centrality values among the nodes,
neither trivially equally spread (delocalization), nor overly focused on a few
nodes (localization). Other reach centralities exhibit both of these behaviors
on regular networks and hub networks, respectively