Network Topology Identification using PCA and its Graph Theoretic Interpretations

We solve the problem of identifying (reconstructing) network topology from steady state network measurements. Concretely, given only a data matrix X where the Xij entry corresponds to flow in edge i in steady-state j, we wish to find a network structure for which flow conservation is obeyed at all the nodes. This models many network problems involving conserved quantities including water, power, metabolic networks, and epidemiology. We show that identification is equivalent to learning a model An which captures the approximate linear relationships between the different variables comprising X (i.e. of the form AnX ≈ 0) such that An is full rank (highest possible) and consistent with a network node-edge incidence structure. We solve this problem through a sequence of steps like estimating approximate linear relationships using Principal Component Analysis, obtaining fundamental cut-sets from these approximate relationships, and graph realization from f-cut-sets (or equivalently f-circuits). Each step and the overall process is polynomial time. The method is illustrated by identifying topology of a water distribution network. We also study the extent of identifiability from steady-state data.