[Paper Review] Economic Networks: Theory and Computation

Thomas J. Sargent, John Stachurski|arXiv (Cornell University)|Mar 22, 2022

Economic theories and models11 citations

TL;DR

This textbook presents a rigorous, quantitative foundation for economic networks using graph theory, linear algebra, and fixed-point methods, with core results centered on the Neumann series lemma and Perron–Frobenius theorem. It integrates spectral theory, optimal flows, Markov chains, and nonlinear dynamics to model production networks, financial contagion, and transport equilibria, offering a unified framework for analyzing systemic risk and network stability in economics and finance.

ABSTRACT

This textbook is an introduction to economic networks, intended for students and researchers in the fields of economics and applied mathematics. The textbook emphasizes quantitative modeling, with the main underlying tools being graph theory, linear algebra, fixed point theory and programming. The text is suitable for a one-semester course, taught either to advanced undergraduate students who are comfortable with linear algebra or to beginning graduate students.

Motivation & Objective

• To establish a mathematically grounded, quantitative framework for economic network analysis, suitable for graduate-level training in economics and applied mathematics.
• To unify key tools from linear algebra, graph theory, and optimization—especially the Neumann series lemma and Perron–Frobenius theorem—for modeling interdependent economic systems.
• To provide a technical foundation enabling researchers to understand and apply network models in general equilibrium, input-output analysis, financial networks, and optimal transport.
• To bridge the gap between introductory network theory and advanced research by offering a self-contained, technically precise treatment of core network concepts with computational implementation in Python and Julia.
• To equip students and researchers with the analytical tools needed to study shock propagation, stability, and equilibrium in multisector and financial networks.

Proposed method

• Uses eigendecompositions and matrix norms to analyze network stability and centrality, particularly via the Perron–Frobenius theorem for nonnegative matrices.
• Applies the Neumann series lemma to derive convergence conditions for iterative processes in input-output and production networks.
• Employs shortest path algorithms and Bellman’s method to compute optimal flows and betweenness centrality in weighted digraphs.
• Integrates linear programming duality and the Kantorovich formulation of optimal transport to model competitive equilibria and resource allocation.
• Models Markov chains on digraphs to study distribution dynamics, ergodicity, and the Markov–Dobrushin coefficient for convergence to stationarity.
• Applies fixed-point theory, especially contraction mappings, to analyze nonlinear systems such as default cascades in financial networks with equity cross-holdings.

Experimental results

Research questions

• RQ1 How do spectral properties of input-output matrices determine the stability and propagation of demand shocks in multisector production networks?
• RQ2 What conditions ensure convergence of iterative processes in economic networks, and how can the Neumann series lemma be used to derive exact stability criteria?
• RQ3 How can optimal transport theory be used to model competitive equilibria and efficient resource flows in networked economies?
• RQ4 In what ways do network structure and centrality measures—such as eigenvector and Katz centrality—predict systemic risk in financial networks?
• RQ5 How do Markov chains on digraphs model information diffusion and social learning, and what role does the Markov–Dobrushin coefficient play in assessing convergence speed?

Key findings

• The Perron–Frobenius theorem ensures the existence of a unique, positive dominant eigenvector for irreducible, nonnegative matrices, enabling the definition of eigenvector centrality in economic networks.
• The Neumann series lemma provides a necessary and sufficient condition for convergence of iterative processes in input-output models, with the spectral radius of the Leontief inverse determining stability.
• Optimal transport problems formulated via the Monge–Kantorovich problem yield competitive equilibria when dual variables are interpreted as shadow prices, linking transport flows to market clearing.
• The Markov–Dobrushin coefficient quantifies the rate of convergence to stationarity in Markov chains on networks, with lower values indicating faster mixing and stronger ergodicity.
• In financial networks, the presence of equity cross-holdings can lead to default cascades that are analytically tractable via fixed-point equations, with contraction mapping conditions ensuring unique solutions.
• The use of vector and matrix norms allows for exact characterization of stability in dynamic network models, with the operator norm serving as a key tool in assessing sensitivity to shocks.