  We provide a framework for determining the centralities of agents in a broad
family of random networks. Current understanding of network centrality is
largely restricted to deterministic settings, but practitioners frequently use
random network models to accommodate data limitations or prove asymptotic
results. Our main theorems show that on large random networks, centrality
measures are close to their expected values with high probability. We
illustrate the economic consequences of these results by presenting three
applications: (1) In network formation models based on community structure
(called stochastic block models), we show network segregation and differences
in community size produce inequality. Benefits from peer effects tend to accrue
disproportionately to bigger and better-connected communities. (2) When link
probabilities depend on geography, we can compute and compare the centralities
of agents in different locations. (3) In models where connections depend on
several independent characteristics, we give a formula that determines
centralities 'characteristic-by-characteristic'. The basic techniques from
these applications, which use the main theorems to reduce questions about
random networks to deterministic calculations, extend to many network games.
