Before revision:  This paper applies Doukhan and Louhichi (1999) 's weak dependence notion to network dependence by measuring dependence strength by the covariance between nonlinearly transformed random variables. While this approach covers examples such as strong mixing random fields on a graph and conditional dependency graphs, it is most useful when dependence arises through a large functional-causal system of equations. The main results of our paper include the law of large numbers , and the central limit theorem  .
After revision:  This paper applies Doukhan and Louhichi (1999) , we measure the strength of dependence by covariances of nonlinearly transformed variables. We provide a law of large numbers , and the central limit theorem .