Reconstructing Large Scale Production Networks

TL;DR

The paper addresses the challenge of reconstructing large-scale, granular production networks from publicly available sectoral flows and firm-size distributions. It introduces a four-step framework: a sector-aware augmented logistic gravity model to generate link probabilities, a Bernoulli backbone to form directed unweighted graphs, a Markov-regularity closure to ensure irreducibility and aperiodicity, and a minimum-energy weighting to produce a stable, weighted network that preserves firm and sector margins. The authors apply the method to the full US economy, reconstructing a network with over 5 million firms and 100 million-plus links, and show that the resulting topology aligns with known properties of real networks and is robust to parameter variation via bootstrap analysis. They also extend the approach to networks with multiple production units (factories) and discuss parallelization and binning strategies that render the method scalable to truly large systems. Overall, the work provides a practical, open-source pipeline for simulating shocks and dynamics on large-scale, granular production networks, enabling policy-relevant analyses that were previously infeasible due to data limitations.

Abstract

This paper develops an algorithm to reconstruct large weighted firm-to-firm networks using information about the size of the firms and sectoral input-output flows. Our algorithm is based on a four-step procedure. We first generate a matrix of probabilities of connections between all firms in the economy using an augmented gravity model embedded in a logistic function that takes firm size as mass. The model is parameterized to allow for the probability of a link between two firms to depend not only on their sizes but also on flows across the sectors to which they belong. We then use a Bernoulli draw to construct a directed but unweighted random graph from the probability distribution generated by the logistic-gravity function. We make the graph aperiodic by adding self-loops and irreducible by adding links between Strongly Connected Components while limiting distortions to sectoral flows. We convert the unweighted network to a weighted network by solving a convex quadratic programming problem that minimizes the Euclidean norm of the weights. The solution preserves the observed firm sizes and sectoral flows within reasonable bounds, while limiting the strength of the self-loops. Computationally, the algorithm is O(N2) in the worst case, but it can be evaluated in O(N) via sector-wise binning of firm sizes, albeit with an approximation error. We implement the algorithm to reconstruct the full US production network with more than 5 million firms and 100 million buyer-seller connections. The reconstructed network exhibits topological properties consistent with small samples of the real US buyer-seller networks, including fat-tails in degree distribution, mild clustering, and near-zero reciprocity. We provide open-source code of the algorithm to enable researchers to reconstruct large-scale granular production networks from publicly available data.

Figures (8)

• Figure 1: Sectoral multiplier parameters $\{\lambda_{l \ell}\}$
• Figure 2: Heat-map of the sectoral multiplier parameter $\{\lambda_{l \ell}\}$ with 24 sectors at the 2-digit SIC level
• Figure 3: Distribution of size elasticity paramerter $\alpha$
• Figure 4: Distribution of sector score exponent parameter $\kappa$
• Figure 5: Distribution of global scaling parameter $z$

Theorems & Definitions (7)

• Remark 1: CLT for the number of edges (density of the network)
• Remark 2: CLT for empirical degree distributions over finitely many bins
• Remark 3: CLT for edge-averaged statistics (e.g., assortativity and reciprocity)
• Remark 4: Stationary money near the empirical firm-size vector
• Remark 5: Finite-sample error bounds for a randomly drawn adjacency matrix
• Remark 6: Minimum-energy as a special case of maximum-entropy
• Remark 7: Error bound on the weighted matrix