Vertex centralities in input-output networks reveal the structure of modern economies

Input-output tables describe the flows of goods and services between the sectors of an economy. These tables can be 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 for such networks. Both are based on random walks and have interpretations as the propagation of supply shocks through the economy. Random walk centrality reveals the vertices most immediately affected by a shock. Counting betweenness identifies the nodes where a shock lingers longest. The two measures differ in how they treat self-loops. We apply both to data from a wide set of countries and uncover salient characteristics of the structures of these national economies. We further validate our indices by clustering according to sectors’ centralities. This analysis reveals geographical proximity and similar developmental status.

Figure 1

(a) This network is taken from Ref. 24. (b) We contrast centrality measures calculated for selected nodes. Even though $c$ is topologically central, our measures do not rank it highly, in contrast to Newman’s betweenness. Instead, they focus on how quickly or how frequently traffic within the network reaches a node. In a graph with two completely connected subcomponents, a slightly remote bridge-like node is not crossed over frequently.

Figure 2

(a) This small network illustrates the importance of a self-loop. (b) The difference between the centrality of node 4 and that of node 3 as a function of the self-loop weight $a_{44}$. All other links have unit weight. Random walk centrality always ranks node 3 highest. Counting betweenness ranks node 4 higher when $a_{44}$ exceeds a threshold near 1.6. If the self-loop has a large weight, it takes a long time before a random walk leaves node 4 and enters the rest of the network.

Figure 3

(a) A hierarchical clustering of countries according to random walk centrality. Gray-scale colors indicate the three important clusters: (1) the industrial countries from Belgium through the United States (black); (2) a mixed group from Argentina through Indonesia, where agriculture and primary products are important (dark gray); and (3) a group of emerging economies from China through Russia (light gray). (b) Clusterings according to counting betweenness. The clusterings according to the two measures are largely stable.