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.