Distributions of Centrality on Networks

TL;DR

This paper develops a framework for analyzing centrality in large random networks, showing that measures are close to their expected values and illustrating economic implications across various network models.

Contribution

It introduces a general method to approximate centrality measures in large random networks, bridging deterministic and stochastic analyses with broad applications.

Findings

Centrality measures concentrate around their expected values in large networks.

Network segregation and community size influence inequality in stochastic block models.

Geography-dependent link probabilities allow for comparative centrality analysis across locations.

Abstract

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.