# Measuring Diffusion over a Large Network

## TL;DR

This paper develops a measure for diffusion of binary outcomes over large networks observed in two periods, identifying causal connections and providing bounds and inference methods, with applications to social networks in India.

## Contribution

It introduces a new causal diffusion measure, identifies conditions for its exactness, and proposes a lower confidence bound with validated inference methods.

## Key findings

- The diffusion measure is identifiable when the observed network contains the causal network.
- A lower confidence bound for diffusion is constructed and shown to be asymptotically valid.
- Monte Carlo simulations confirm the method's finite sample stability.

## Abstract

This paper introduces a measure of the diffusion of binary outcomes over a large, sparse network, when the diffusion is observed in two time periods. The measure captures the aggregated spillover effect of the state-switches in the initial period on their neighbors' outcomes in the second period. This paper introduces a causal network that captures the causal connections among the cross-sectional units over the two periods. It shows that when the researcher's observed network contains the causal network as a subgraph, the measure of diffusion is identified as a simple, spatio-temporal dependence measure of observed outcomes. When the observed network does not satisfy this condition, but the spillover effect is nonnegative, the spatio-temporal dependence measure serves as a lower bound for diffusion. Using this, a lower confidence bound for diffusion is proposed and its asymptotic validity is established. The Monte Carlo simulation studies demonstrate the finite sample stability of the inference across a range of network configurations. The paper applies the method to data on Indian villages to measure the diffusion of microfinancing decisions over households' social networks.