In many social and economic networks, agents' outcomes depend substantially on the centrality of their network position. Our current understanding of network centrality is largely restricted to deterministic settings, but in many applications data limitations or theoretical concerns lead practitioners to use random network models. We provide a foundation for understanding how central agents in random networks are likely to be. Our main theorems show that on large random networks, centrality measures are close to their expected values with high probability. By applying these theorems to stochastic block models, we study how segregated networks contribute to inequality. When networks are segregated, benefits from peer effects tend to accrue unevenly to the advantage of more central individuals and groups. We also discuss applications to more general network formation models, including models where link probabilities are governed by geography.
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