We analytically derive the updates of a transmission-line network's interaction matrix and scattering matrix as a consequence of a fault (an interrupted transmission line). We find that the fault alters not only the direct coupling between the two nodes that were previously connected by the faulty cable, but that the fault also alters these nodes' self-interactions in a non-trivial manner. Given the network's topology, it is then possible to remotely localize the fault on the faulty cable based on measurements of the faulty network's scattering coefficient(s). Our analytical expressions make it possible to efficiently calculate the expected scattering matrix for different fault locations (orders of magnitude faster than a brute-force evaluation). We report a simple demonstration for which we assume to know the network's topology as well as which cable is faulty; we identify the location of the fault on the faulty cable by comparing the broadband scattering coefficient(s) swept across candidate fault locations to the one(s) measured on the faulty network.