Preprocessing graphs for network inference applications

The problem of network inference can be solved as a constrained matrix factorization problem where some sparsity constraints are imposed on one of the matrix factors. The solution is unique up to a scaling factor when certain rank conditions are imposed on both the matrix factors. Two key issues in factorising a matrix of data from some netwrok are that of establishing simple identifiability conditions and decomposing a network into identifiable subnetworks. This paper solves both the problems by introducing the notion of an ordered matching in a bipartite graphs. Novel and simple graph theoretical conditions are developed which can replace the aforementioned computationally intensive rank conditions. A simple algorithm to reduce a bipartite graph and a graph preprocessing algorithm to decompose a network into a set of identifiable subsystems is proposed.