Input-Output Networks

Vertex Centralities in Input-Output Networks


In empirical economics, input-output tables describe the flows of goods and services between the different sectors of an economy. These tables can be immediately interpreted as weighted directed networks. At the usual level of aggregation, they contain nodes with strong self-loops and are almost completely connected. We derive two measures of node centrality that are well suited such networks. Both are based on random walks and have a direct economic interpretation in the propagation of shocks through an economy. 

Random walk centrality reveals the vertex most immediately affected by a shock. Our second measure called counting betweenness identifies the nodes where a shock lingers longest. We demonstrate that our measures differ in their reaction to self-loops. We apply both measures to data from a wide set of countries. They uncover salient characteristics of the structure of the national economies. To further validate our indices, we perform a clustering of the sectors' centralities, which reveals geographical proximity and similar developmental status.


MATLAB code for the centrality measures: 

Random walk centrality.m; this file needs  mfpt.m

Counting betweenness.m

We provide the OECD data as a MATLAB workspace:


The following EXCEL sheets contain the complete node rankings for all countries. 

Rankings according to random walk centrality

Rankings according to counting centrality



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Blöchl F, Theis FJ, Vega-Redondo F, and Fisher E: Vertex Centralities in Input-Output Networks Reveal the Structure of Modern Economies, Physical Review E, 83(4):046127, 2011. Link

Blöchl F, Theis FJ, Vega-Redondo F, and Fisher E: Which sectors of a modern economy are most central?, CESifo Working Paper Series No. 3175. Link