  We study strategic network formation games in which agents attempt to form
(costly) links in order to maximize their network centrality. Our model derives
from Jackson and Wolinsky's symmetric connection model, but allows for
heterogeneity in agent utilities by replacing decay centrality (implicit in the
Jackson-Wolinsky model) by a variety of classical centrality and game-theoretic
measures of centrality. We are primarily interested in characterizing the
asymptotically pairwise stable networks, i.e. those networks that are pairwise
stable for all sufficiently small, positive edge costs. We uncover a rich
typology of stability:
  - we give an axiomatic approach to network centrality that allows us to
predict the stable network for a rich set of combination of centrality utility
functions, yielding stable networks with features reminiscent of structural
properties such as "core periphery" and "rich club" networks.
  - We show that a simple variation on the model renders it universal, i.e.
every network may be a stable network.
  - We also show that often we can infer a significant amount about agent
utilities from the structure of stable networks.
