Distributions of centrality on networks

We provide a framework for determining agents' centralities 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. 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 via three applications: (1) In network formation models with community structure, we show network segregation and differences in community size produce inequality. Benefits from peer effects accrue disproportionately to bigger and better-connected communities. (2) When link probabilities depend on spatial structure, we compute and compare the centralities of agents in different locations. (3) In models where connections depend on several independent characteristics, we can determine 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.