Causal and non-causal filtering for network reconstruction

In many diverse areas, determining the connectivity of various entities in a network is of significant interest. This article´s main focus is on a network (graph) of nodes (vertices) that are linked via filters that represent the edges of a graph. Both cases of the links being non-causal and causal are considered. Output of each node of the graph represents a scalar stochastic process driven by an independent noise source and by a sum of filtered outputs of nodes linked to the node of interest. It is shown that the method provided will identify all true links in the network with some spurious links added. The spurious links remain local in the sense that they are added within a hop of a true link. In particular, it is proven that the method determines a link to be present only between the kins of a node where kins of a node consist of parents, children and co-parents (other parents of all of its children) in the graph. Main tools for determining the network topology is based on Wiener filtering. Another significant insight provided by the article is that the Wiener filter estimating a stochastic process, represented by a node, based on other processes in a network configuration remains local in the sense that the Wiener filter utilizes only measurements local to the node being estimated.