Deriving production chains using restricted gradient extraction

So as to assess systemic economic value and risk, it is of crucial interest to develop methods to detect interwoven production chains in large-scale economies. Commodity transactions between firms induce a complex network in which it is challenging to identify production chains, not only due to the size of the underlying network but also because of its inherent cyclical connections and loops. We present a novel method, Restricted Gradient Extraction (RGE), which is based on Hodge decomposition, which is capable of extracting the gradient flow of a production network. The RGE method, being of relatively low computational complexity, is demonstrated to both a synthetic and real country-sized production network. Application of RGE on the syntactic data set shows that the resulting gradient flow is a directed acyclic graph, a weighted subgraph of the original network, and the gradient flow is retained. Application to the economic production network of the Netherlands shows that production chains can be readily detected and described. The method is applicable to weighted directed networks in general and is not limited to economic production networks.