A New Method for Reconstructing Network Topology from Flux Measurements

In this paper, we solve the problem of reconstructing or identifying the network topology from steady state flux measurements. Specifically, given a data matrix $\mathbf{X}$, where each column contains measurements corresponding to a distinct steady state configuration, we wish to estimate a model $\mathbf{A}$ which captures the approximate linear relationships between different variables comprising $\mathbf{X}$ (i.e. of the form $\mathbf{AX \approx 0}$) such that $\mathbf{A}$ is full rank (highest possible) and consistent with a network incidence structure. Hence the network topology can be obtained just by looking at this estimated model. The problem is solved through a sequence of steps including estimating approximate linear relationships between variables using Principal Component Analysis (PCA), obtaining fundamental cut-sets from the model estimated by PCA, and graph realization from f-cut-sets (or equivalently f-circuits). Each step and the overall process is polynomial time. The method is illustrated using an example of water network reconstruction using steady-state flow rate measurements. We also establish hard limits on what can be, and cannot be done with steady-state data.