  This paper proposes a family of network centralities called fixed-point
centralities. This centrality family is defined via the fixed point of
permutation equivariant mappings related to the underlying network. Such a
centrality notion is immediately extended to define fixed-point centralities
for infinite graphs characterized by graphons. Variation bounds of such
centralities with respect to the variations of the underlying graphs and
graphons under mild assumptions are established. Fixed-point centralities
connect with a variety of different models on networks including graph neural
networks, static and dynamic games on networks, and Markov decision processes.
