# Testing for Differences in Stochastic Network Structure

## TL;DR

This paper develops new statistical tests to detect whether treatments or shocks change the structure of social networks, demonstrating that certain tests are more powerful for typical economic networks.

## Contribution

It introduces network-specific two-sample tests using operator norms, extending the Kolmogorov-Smirnov approach to network data with proven power advantages.

## Key findings

- The $\infty o1$ norm test outperforms the $2 o2$ norm test in sparse networks.
- The tests are validated through analytical, simulation, and real-world data.
- The methods effectively detect treatment-induced changes in network structure.

## Abstract

How can one determine whether a community-level treatment, such as the introduction of a social program or trade shock, alters agents' incentives to form links in a network? This paper proposes analogues of a two-sample Kolmogorov-Smirnov test, widely used in the literature to test the null hypothesis of "no treatment effects", for network data. It first specifies a testing problem in which the null hypothesis is that two networks are drawn from the same random graph model. It then describes two randomization tests based on the magnitude of the difference between the networks' adjacency matrices as measured by the $2\to2$ and $\infty\to1$ operator norms. Power properties of the tests are examined analytically, in simulation, and through two real-world applications. A key finding is that the test based on the $\infty\to1$ norm can be substantially more powerful than that based on the $2\to2$ norm for the kinds of sparse and degree-heterogeneous networks common in economics.