Effects of hidden nodes on noisy network dynamics

We consider coupled network dynamics under uncorrelated noises, but only a subset of the network and their node dynamics can be observed. The effects of hidden nodes on the dynamics of the observed nodes can be viewed as having an extra effective noise acting on the observed nodes. These effective noises possess spatial and temporal correlations whose properties are related to the hidden connections. The spatial and temporal correlations of these effective noises are analyzed analytically and the results are verified by simulations on undirected and directed weighted random networks and small-world networks. Furthermore, by exploiting the network reconstruction relation for the observed network noisy dynamics, we propose a scheme to infer information of the effects of the hidden nodes such as the total number of hidden nodes and the weighted total hidden connections on each observed node. The accuracy of these results are demonstrated by explicit simulations.