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.

翻译：本文提出了一种度量二元结果在大型稀疏网络上扩散程度的方法，前提是该扩散现象在两个时间段内被观测到。该度量捕捉了初始时段内状态转换对其邻居在第二个时间段结果所产生的聚合溢出效应。本文引入了一个因果网络，用以刻画跨两个时间段的横截面单元之间的因果联系。研究表明，当研究者观测到的网络包含该因果网络作为子图时，扩散度量可被识别为观测结果的一种简单时空依赖度量。当观测网络不满足此条件，但溢出效应非负时，该时空依赖度量可作为扩散的下界。基于此，本文提出了扩散的下置信界，并证明了其渐近有效性。蒙特卡洛模拟研究展示了该推断方法在不同网络配置下的有限样本稳定性。本文将该方法应用于印度村庄的数据，以衡量小额信贷决策在家庭社交网络上的扩散。