  We consider several families of network centrality measures induced by graph
kernels, which include some well-known measures and many new ones. The
Self-consistency and Bridge axioms, which appeared earlier in the literature,
are closely related to certain kernels and one of the families. We obtain a
necessary and sufficient condition for Self-consistency, a sufficient condition
for the Bridge axiom, indicate specific measures that satisfy these axioms, and
show that under some additional conditions they are incompatible. PageRank
centrality applied to undirected networks violates most conditions under study
and has a property that according to some authors is ``hard to imagine'' for a
centrality measure. We explain this phenomenon. Adopting the Self-consistency
or Bridge axiom leads to a drastic reduction in survey time in the culling
method designed to select the most appropriate centrality measures.
