AMatrix-Algebraic Framework and Pairwise Asymmetry Multiplier for Quantifying Shock Propagation in Complex Production Networks

Scalar centrality measures assign one score to each sector of a production network and cannot compare its exposure to two shock sources. We develop a matrix-algebraic framework for this comparison. The Asymmetry Multiplier ${\gamma }_j(\alpha ,\beta )=L_{j\alpha }^{\top }/L_{j\beta }^{\top }$, the ratio of the price responses of a target sector to equal increments in unit value added at two sources, factors into a direct ratio and a network amplification ratio. Using sub-eigenvectors of the technical coefficient matrix, we derive lower and upper bounds that hold for every productive network and reduce to Perron–Frobenius bounds under irreducibility. We also prove that linear price responses bound those of competitive constant-returns economies from above and that no labeling-independent node-level measure determines the multiplier. A symmetric structural decomposition separates technological and final demand effects. As a proof of concept, the framework is applied to electricity and mining cost shocks in Türkiye. In 2022, the multiplier ranges from 0.20 to 1.86 across eleven sectors, indirect paths attenuate the direct ratio by a factor of 8 to 22 in six sectors, and Basic Metals is more exposed to mining than to electricity cost shocks, through imported mining inputs. For most sectors, the dominant channel is robust to disaggregation, coefficient perturbation and the annual tables of 2012–2022.