A Markovian model of evolving world input-output network

Abstract

The initial theoretical connections between Leontief input-output models and Markov chains were established back in 1950s. However, considering the wide variety of mathematical properties of Markov chains, so far there has not been a full investigation of evolving world economic networks with Markov chain formalism. In this work, using the recently available world input-output database, we investigated the evolution of the world economic network from 1995 to 2011 through analysis of a time series of finite Markov chains. We assessed different aspects of this evolving system via different known properties of the Markov chains such as mixing time, Kemeny constant, steady state probabilities and perturbation analysis of the transition matrices. First, we showed how the time series of mixing times and Kemeny constants could be used as an aggregate index of globalization. Next, we focused on the steady state probabilities as a measure of structural power of the economies that are comparable to GDP shares of economies as the traditional index of economies welfare. Further, we introduced two measures of systemic risk, called systemic influence and systemic fragility, where the former is the ratio of number of influenced nodes to the total number of nodes, caused by a shock in the activity of a node, and the latter is based on the number of times a specific economic node is affected by a shock in the activity of any of the other nodes. Finally, focusing on Kemeny constant as a global indicator of monetary flow across the network, we showed that there is a paradoxical effect of a change in activity levels of economic nodes on the overall flow of the world economic network. While the economic slowdown of the majority of nodes with high structural power results to a slower average monetary flow over the network, there are some nodes, where their slowdowns improve the overall quality of the network in terms of connectivity and the average flow of the money.

Fig 1. A schematic view of a closed economic network.

There are two economies and one industry within each economy, and one node for the government and households within each economy. The edges represent the flow of money between nodes.

Fig 2. The sequence of average mixing time of Markov chains as an aggregate index of globalization.

Lower values indicate more globally connected network. The error bars represent 3 standard deviations.

Fig 3. The sequence of Kemeny constants of Markov chains as an aggregate index of globalization.

Lower values indicate more globally connected networks.

Fig 4. GDP shares of economies (red line) compared with their aggregated structural powers (blue line) over time.

The ratio between two time series reveals the structural potential of the economies for further growth (blue gaps) or the risk of economic failure (red gaps).

Fig 5. Predicted trends of structural potential of different economies.

Fig 6. The effect of 99% slowdown of electrical and optical equipment industry of China.

Left side shows the shocked network in 1995 the right side shows 2011. The green (red) color declares an increase (decrease) in the final share (structural power, π_t,i) of the node as a result of the slow down in the selected industry.

Fig 7. Systemic fragility vs. systemic influence of each industry for the year 2011.

Fig 8. Effect of slowing downs the activity of economic nodes on Kemeny constant in 2011.

Fig 9. The paradoxical effect of slowdown in the activity economic nodes (except rest of the world) on the Kemeny constants.