# Ranking the Economic Importance of Countries and Industries

Wei Li  Center for Polymer Studies and Department of Physics, Boston
University, Boston, USA  Dror Y. Kenett  Center for Polymer Studies and
Department of Physics, Boston University, Boston, USA  Kazuko Yamasaki
Department of Environmental Sciences, Tokyo University of Information
Sciences, Japan  H. Eugene Stanley  Center for Polymer Studies and Department
of Physics, Boston University, Boston, USA  Shlomo Havlin  Department of
Physics, Bar-Ilan University, Israel

###### Abstract

In the current era of worldwide stock market interdependencies, the global
financial village has become increasingly vulnerable to systemic collapse. The
recent global financial crisis has highlighted the necessity of understanding
and quantifying interdependencies among the world’s economies, developing new
effective approaches to risk evaluation, and providing mitigating solutions.
We present a methodological framework for quantifying interdependencies in the
global market and for evaluating risk levels in the world-wide financial
network. The resulting information will enable policy and decision makers to
better measure, understand, and maintain financial stability. We use the
methodology to rank the economic importance of each industry and country
according to the global damage that would result from their failure. Our
quantitative results shed new light on China’s increasing economic dominance
over other economies, including that of the USA, to the global economy.

##  I Introduction

The growth of technology, globalization, and urbanization has caused world-
wide human social and economic activities to become increasingly
interdependent helbing2010fundamental ; havlin2012challenges ;
san2012challenges ; helbing2012systemic ; lazer2009life ; king2011ensuring ;
lorenz2011social ; yamasaki2012complex ; meng2009production ;
rinaldi2001identifying ; solomon2003pioneers ; levy2010scale ;
klimek2012empirical . From the recent financial crisis it is clear that
components of this complex system have become increasingly susceptible to
collapse. Integrated models, currently in use, have been unable to predict
instability, provide scenarios for future stability, or control or even
mitigate systemic failure. Thus, there is a need of new ways of quantifying
complex system vulnerabilities as well as new strategies for mitigating
systemic damage and increasing system resiliency farmer2009economy ;
lux2009economics . Achieving this would also provide new insight into such key
issues as financial contagion forbes2001measuring ; forbes2002no  and systemic
risk bodie2002investments ; billio2010econometric ; bisias2012survey  and
would provide a way of maintaining economic and financial stability in the
future.

It is clear from the recent crisis that different sectors of the economy are
strongly interdependent. The housing bubble in the USA caused a liquidity
freeze in the international banking system, which in turn triggered a massive
slowdown of the real economy costing many trillions of dollars and threatening
the financial integrity of the European Union. This demonstrates the high
level of dependence between different components in the world economic system.
Because strong nonlinearities and feedback loops make economic systems highly
vulnerable, we need to understand the behavior of the interacting networks
that comprise the economy. How do they interact with each other, and what are
their vulnerabilities? One may consider an industry in two countries
respectively, for example, electrical equipment industry in China and the USA,
and ask which industry is more important to economic stability, electrical
equipment in China or that in the USA? Is electrical equipment in China more
critical for global economy stability when the production in electrical
equipment in China is reduced, and how it will impact the industries within
the country and industries abroad.

To answer these questions, we employ recent advances in the theory of
cascading failures in interdependent networks buldyrev2010catastrophic ;
gao2011networks ; li2012cascading . In the case of interdependent networks, a
malfunction of only a few components can lead to cascading failures and to a
sudden collapse of the entire system. This is in contrast to single isolated
networks, which tend to collapse gradually parshani2010interdependent . These
recent results indicate the central importance of interconnectivity and
interdependency to the stability of the entire system. There have been studies
of the complex set of coupled economic networks garas2010worldwide ;
schweitzer2009economic ; huang2013cascading ; hidalgo2008network . However,
the importance of countries and industries in the stability of the global
economy have not been analyzed, and there is a need for useful methods to
quantify and rank their economic importance and influence.

The input-output (IO) model is a technique that quantifies interdependency in
interconnected economic systems. Wassily Leontief leontief2004wassily  first
introduced the IO model in 1951, for which he received the Nobel Prize in
Economics in 1973. It can be used to study the effect of consumption shocks on
interdependent economic systems isard1960methods ; lahr2001input ;
ten2005economics ; miller2009input ; santos2006inoperability ;
pokrovskii2012econodynamics ; leontiev1986input . Analysis of IO data is
performed using such techniques as the hypothetical extraction method (HEM)
miller2001taxonomy ; temurshoev2010identifying . Although HEM can measure the
relative stimulative importance of a given industry by calculating output with
and without the industry being examined, it does not quantify each industry’s
full spectrum of importance to the stability of global economic system. For
example, if an industry in a given country collapses completely due to natural
disaster or civil unrest it will no longer be able to consume products
supplied by other industries. This can cause cascading failure in the economic
system if the other industries cannot function when the cash flow from the
failed industry is removed. By measuring how widely the damage spreads, we
will rank here an industry’s importance within the world-wide economic system.

In this paper we examine the interdependent nature of economies between and
within 14 countries and the rest of the world (ROW). We use an “input-output
table” timmer2012world  and focus on economic activity during the period
1995–2011. From the table we analyze data from 14 countries (Australia,
Brazil, Canada, China, Germany, Spain, France, UK, India, Italy, Japan, Korea,
Russia, and USA) and from the rest of the world (ROW). The economic activity
in each country is divided into 35 industrial classifications. Each cell in
the table shows the output composition of each industry to all other 525
industries and its final demand and export to the rest of the world (see Ref.
wiotdata ). We construct an output network using the 525 industries as nodes
and the output product values as weighted links based on the input-output
table, and focus on the output product value for each industry.

Our goal is to introduce a methodology for quantifying the importance of a
given industry in a given country to global economic stability with respect to
other industries in countries that are related to this industry. Thus, we
study the inflow and outflow of money between each set of 35 industries and
the ROW in each of the 14 countries (see the Data section for more detail). We
use the theory of cascading failures in interdependent networks to gain
valuable information on the local and global influence on global stability of
different economic industries, a methodology that can provide valuable new
insights and information to present-day policy and decision makers.

##  II World Input-Output Table Data

The database we use in this study is the World Input-Output Table (WIOT)
wiotdata . It provides data for 27 European countries, 13 other countries, and
the rest of the world (ROW) for the period 1995–2011. For our sample, we
select the 14 countries with the largest domestic input and the largest import
in production in 2011 and also the ROW (adding the remaining 26 countries into
the ROW). Using this sample, we construct a new input-output industry-by-
industry table. For sake of simplicity, we assume that each industry produces
only one unique product. In the WIOT, the column entries represent an
industry’s inputs and the row entries represent an industry’s outputs. The
rows in the upper sections indicate the intermediate or final use of products.
A product is intermediate when it is used in the production of other products
(intermediate use). The final use category includes domestic use (private or
government consumption and investment) and exports. The final element in each
row indicates the total use of each product. The industry columns in the WIOT
contain information on the supply of each product. The columns indicate the
values of all intermediate, labor, and capital inputs used in production.
Total supply of the product in the economy is determined by domestic input
plus final demand. The IO table contains negative numbers as outputs for
various reasons timmer2012world , but their fraction is fairly small and their
values are small as well. For simplicity, we set these negative numbers as
zero in our analysis.

Based on the supply of the product for each industry, it is possible to
construct a directed product supply network. Then we reverse the direction of
the links in the network (see schematic representation in Fig. 1), and the
network represents the money outflow from one industry to another; e.g.,
electrical equipment industry is pointing to machinery industry because
electrical equipment industry buys a product from machinery industry. The
links are weighted according to the value of products from machinery industry
as its output electrical equipment industry as its input.

##  III Industry tolerance

In order to quantify and rank the influence of industries in the stability of
this global network, we perform a cascading failure tolerance analysis
buldyrev2010catastrophic . Our model is described as follows. Suppose industry
A fails, other industries can no longer sell their products to industry A and
thus they lose that revenue. The revenue industry B is reduced by a fraction
$p^{\prime}$, which is defined as industry B’s revenue reduction caused by the
failure of industry A divided by that industry B’s total revenue. The
tolerance fraction $p$ is the threshold above which an industry fails. This
occurs when reduced revenue fraction $p^{\prime}$ is larger than tolerance
fraction $p$. Here we assume that (i) $p$ is the same for all industries and
that every industry fails when its $p^{\prime}>p$ and (ii) the failure of an
industry in country A does not reduce the revenue of the other industries in
the same country A because they are able to quickly adjust to the change.

The methodology can be schematically illustrated as follows: In step 1,
industry $A$ in country $i$ fails. This causes other industries in other
countries to fail if their $p^{\prime}>p$. Assume that in step 2 industries
$B$, $C$, and $D$ fail. The failure of these industries in step 2 will reduce
other industries’ revenue and cause more industries including those in country
$i$ to have a reduced fraction $p^{\prime}$. Thus in step 3 there is an
increased number of industries whose $p^{\prime}>p$. Eventually the system
reaches a steady state in which no more industries fail. The surviving
industries will all have a reduced revenue fraction that is smaller than the
tolerance fraction, i.e., $p^{\prime}\leq p$. Figure 1 demonstrates this
process, and the subfigures show the steps in the cascading failure.

Figure 1: Schematic representation of each step in the cascading failure
propagation in the world economic network. We present an example of two
countries, where circle nodes represent country 1, and triangles represent
country 2. Both countries have the same industries, and the arrow between two
nodes points in the direction of money flow. (a) There is a failure in
electrical equipment industry in Country 1 (circle) which causes a failure of
electrical equipment industry in Country 2. In (b) the failure of electrical
equipment industry in Country 2 will cause rubber and plastics, wholesale,
finance and chemicals in Country 2 to fail. (c) Shows mining in Country 2 and
chemicals in Country 1 to fail further. (d) The network reached a steady
state. The red nodes represent the failing industries and the yellow nodes
stand for the surviving industries.

To determine how much the failure of each industry would impact the stability
of the economic network, we change the tolerance fraction $p$ from 0 to 1 and
measure the fraction of surviving industries left in the network. When the
tolerance fraction approaches 0, any revenue reduction caused by the failure
of one industry can easily destroy almost all the other industries in the
network, and the network will collapse. When the tolerance fraction approaches
1, all the industries can sustain a large reduction of revenue and the failure
of one industry will not affect the others.

Figure 2(a) shows the failures of electrical equipment industry in China and
energy industry in the United States for the 2009 WIOT and shows the fraction
of the largest cluster of connected and functional industries as a function of
the tolerance fraction $p$ after the Chinese electrical equipment industry
becomes malfunction and is removed from the network due to a large shock to
this industry. The shock could result from different causes, such as natural
environmental disaster, government policy changes, insufficient financial
capability. The removal of China electrical equipment industry will cause
revenue reduction in other industries because China electrical equipment
industry is not able to buy products and provide money to other industries.
When $p$ is small, the industries are fragile and sensitive to the revenue
reduction, causing most of the industries fail, and the number of the
surviving industries is very small. When $p$ is large, the industries can
tolerate large revenue reduction and are more robust when revenue decreases.
The number of the surviving industries tends to increase abruptly at a certain
$p=p_{c}$ value as $p$ increases. Figure 2(b) shows the number of cascading
steps that elapse before a stable state is reached as a function of tolerance
fraction $p$ after removing the Chinese electrical equipment industry or the
USA energy industry. The number of steps reaches a peak when $p$ approaches
criticality $p_{c}$parshani2010interdependent .

Figure 2: Typical examples of industry tolerance threshold $p_{c}$. Tolerance
threshold $p_{c}$ shows the importance of each industry in the international
industry networks. In (a) the black curve shows the fraction of surviving
industries as a function of tolerance threshold for the case when the
electrical equipment industry in China fails in year 2009 and the red curve
represents the case of the failure of energy industry in 2009 in the USA. (b)
Shows the number of failure steps as a function of $p$ corresponding to (a).
The total number of steps is the number of cascades it takes for the network
to reach a steady state after certain initial failure. The peaks in (b) are
corresponding to the abrupt jumps in (a), which means that large numbers of
cascade steps are associated at $p_{c}$ with a dramatic failures in the
industry network. Each step in the cascading process in (b) is demonstrated in
Figure 1.

Here we analyze for each of the 525 industries this important parameter, the
critical tolerance threshold $p_{c}$. Our goal is to determine the threshold
$p_{c}$ at which global network subject to collapse becomes stable and most of
the industries in the network survive after initial failures. To this end, we
assume that the $p_{c}$ is the critical threshold below which less than 30% of
industries survive. When $p>p_{c}$, more than 30% of the remaining industries
survive after the failure cascade in the system is over. When $p<p_{c}$, the
survival rate of the remaining industries is 30% or less. The higher this
threshold, the higher the impact of a failing industry will be on the
vulnerability of the global network. Without losing generalization, we also
take different fraction of surviving industries to set $p_{c}$ in simulations.
For example, when the $p_{c}$ is the threshold below which 50% of industries
survive, the importance of industries and countries maps are shown as in
supplementary information. The correlation of $p_{c}$ values in simulations
using 50% fraction of surviving industries and 30% surviving industries is 1
and the $p_{c}$ are identical in both scenarios. Thus, we find that the
critical threshold $p_{c}$ is not sensitive to the chosen fraction in this
industry network, for values below 50%. The following $p_{c}$ in this study is
defined as the critical threshold below which less than 30% of industries
survive.

We use this methodology to test how the failure of an individual industry in a
given country impacts the stability of the entire system. Thus, $p_{c}$ of an
industry is our measure of the importance of this industry in the global
economic network.

Using the tolerance threshold $p_{c}$, we can quantify and rank the economic
importance of each industry. We measure the tolerance threshold of 35
industries in 14 countries between 1995 and 2011. We calculate the tolerance
of each industry according to how much it affects the entire network, i.e.,
all 35 industries in all 14 countries.

##  IV Importance of Country and Industry

The proposed methodology provides the means to quantify and rank the
importance of each industry to the stability of the global economic network or
the importance of each country in the global economic network.

We define the importance of each country by averaging all the 35 industries
$p_{c}$ within the country for a specific year,

$I_{country}(i)\equiv\frac{1}{n}\sum_{k=1}^{n}{p_{c}(i,k)}$ (1)

where $n$ is the number of industries. To rank the importance of a given
country, we average the tolerance across industries as shown in Eq. 1.

Figure 3(a) shows the importance of all countries for different years where
the average in Eq. 1 is taken over the four largest $p_{c}$ values, in order
to consider the strongest industries. Note that the rest of the world (RoW) is
the most important ”country” since it includes many countries other than the
14 countries included in the sample. Countries in RoW provide products that
are crucial inputs to these 14 countries and their industries. We also define
the importance of the individual industries in terms of the average of their
$p_{c}$,

$I_{industry}(i)\equiv\frac{1}{T}\sum_{j=1}^{T}{p_{c}(i,t)},$ (2)

where $T$ denotes years. Figure 3(b) shows the average tolerance of industry
over all years (1995-2011) for the individual countries. For the sake of
clarity, we plot only the top 20 industries with respect to the average of
$I_{\rm industry}$ values for all the countries. Note that the electrical
equipment industry is the most important when we average the tolerance
fraction $p_{c}$ across 17 years. The energy industry $p_{c}$ in the United
States, for example, is relatively high because the United States is the
world’s largest energy consumer.

Figure 3: Ranking the importance of individual industries or countries. (a)
Map of averaging industry tolerances for each country, $I_{country}$. The
average tolerance $p_{c}$ is calculated by averaging the largest four $p_{c}$
for each country in one year. The color represents the strength of the
tolerance, ranging from blue for low tolerance, to red for high tolerance. (b)
Averaging industry tolerance over 17 years (1995-2011). The average tolerance
$p_{c}$ is calculated by averaging the industry tolerance over 17 years for
each country (for presentation sake, we plot only the top 20 industries with
respect to their $I_{industry}$ value). The values are represented using the
same color code.

##  V Robustness and stability of economic structure

We use the critical tolerance threshold $p_{c}$ to measure the importance of
each industry in the global economic network. We can also rank each industry
according to their tolerance $p_{c}(i,k)$ for each separate country—the
tolerance of industry $i$ in country $k$. By comparing industry order rankings
for different years, we can study the similarity in economic environment
across a period of 17 years.

To do this we use the Kendall $\tau$ coefficient kendall1938new , which
measures rank correlation, i.e., the similarity in data orderings when ranked
by each quantity. Let $(x_{1},y_{1}),(x_{2},y_{2}),\dots,(x_{n},y_{n})$ be a
set of observations of the random variables $X$ and $Y$ respectively. Any pair
of observations $(x_{i})$ and $(y_{i})$ is concordant if the ranks for both
elements agree, that is if $x_{i}<x_{j}$ and $y_{i}<y_{j}$ or if $x_{i}>x_{j}$
and $y_{i}>y_{j}$. Otherwise the pair is discordant. If $x_{i}=x_{j}$ or
$y_{i}=y_{j}$, the pair is neither concordant nor discordant. The Kendall
$\tau$ coefficient is defined as

$\tau=\frac{n_{+}-n_{-}}{\frac{1}{2}n(n-1)},$ (3)

where $n_{+}$ is the number of concordant pairs and $n_{-}$ is the number of
discordant pairs. The coefficient $\tau$ is in the range $-1\leq\tau\leq 1$.
When the agreement between the two rankings is perfect, the coefficient is 1.
When the disagreement between the two rankings is perfect, the coefficient is
$-1$. If $X$ and $Y$ are independent, the coefficient is approximately zero.

(a) China

(b) USA

(c) Germany

Figure 4: Kendall correlation coefficient heat map shows the industry rank
correlation coefficient between each pair of years from $1995-2011$. Using
Kendall correlation coefficient $\tau$, we investigate the evolution of the
economic structure of the investigated countries. For each year, we rank the
industries in each country according to their tolerance values, and then
calculate the Kendall $\tau$ for every pairs of years. We plot the values of
the Kendall $\tau$ for all pairs of years, for China (top), USA (middle), and
Germany (bottom), using a color code ranging from blue, for low similarity, to
red, for high similarity (see Supplementary Information for all other
countries).

We use the Kendall $\tau$ to investigate the evolution of the economic
structure of each country in our sample. For each year, we rank the industries
in each country according to their tolerance values, and calculate the Kendall
$\tau$ for every year pair. Figure 4 shows the values of the Kendall $\tau$
for all pairs of years, for China (Fig. 4(a)), USA (Fig. 4(b)), and Germany
(Fig.4(c)), using a color code ranging from blue, for low similarity, to red,
for high similarity (see Supplementary Information for all other countries).
For these three countries, we find different behaviors in terms of the
stability and consistency of the economic structures. For the case of China,
we observe that the structure changes significantly, with high values of the
Kendal correlation (represented using red) only presenting for previous
2$\sim$3 years (as can be observed from the diagonal of elements of Fig.
4(a)). However, in the case of the USA, it is possible to observe three
periods in terms of stability of the economic structure: 1995-1999, 2000-2007,
and 2008-2011 (see Fig. 4(b)). The first marks the period leading to the
”dot.com” crisis, which was followed by a significant change in the US market
structure. The second marks the period leading unto the financial crisis,
which again was followed by a significant change in the US market structure.
Finally, the third marks the period following the financial crisis, which
shows no stable period in terms of market structure. Finally, in the case of
Germany, we observe only two periods: 1995-1999, and 2000-2011 (see Fig.
4(c)). The first period can be also attributed to that apparent in the case of
the USA, however possibly to a lesser extent and also resulting from the
introduction of the Euro currency. However, in the case of Germany we do not
observe a change in structure following the recent financial crisis, which
highlights the degree of stability in the German economy.

##  VI The rise of China

Due to the fact that economic influence is dynamic across time, we ask whether
the methodology presented here can provide new information on the increase or
decline of economic importance. In each year we calculate the individual
industry tolerance as described above. We then calculate the average tolerance
of each country for a given year. Figure 5 shows the average of $p_{c}$ of
country for the 17-year period investigated. In Figure 5(a), we show the
largest tolerance $p_{c}$ in China, the USA and Germany along 17 years. Figure
5(b) shows the average of 4 largest industries $p_{c}$, and Figure 5(c) shows
the average of 8 largest $p_{c}$ in each country.

We find that the average tolerance $p_{c}$ of China becomes larger than that
of the USA after 2003, which is most pronounced in Figure 5(b). The USA
tolerance $p_{c}$ first increases from 1995 to 2000, then decreases from 2000
to 2009, with a slight increase in 2010-2011. Germany’s tolerance $p_{c}$ in
general increase in the investigated period and shows certain fluctuations
between 2000 and 2005. Note that for the USA the change across time is minor
but that the economic importance of China increases significantly. The
economic importance of China relative to that of the USA shows a consistent
increase from year to year, illustrating how the economic power structure in
the world’s economy has been changing during time.

Figure 5: Tolerance $p_{c}$ changes of China, the USA and Germany for 17
years. We show that the average tolerance $p_{c}$ increases from 1995 to
2011\. In (a), we shows the largest tolerance $p_{c}$ in China, the USA and
Germany along 17 years. (b) Shows the average of 4 largest $p_{c}$ in each
country. We can see the $p_{c}$ of China becomes larger than the USA after
2003. (c) Shows the average of 8 largest $p_{c}$ in each country. The
difference between the average $p_{c}$ of China and that of the USA is smaller
compared to that in (b) in years 2005-2011. This is because the average
includes more industries with small $p_{c}$ and mainly the importance of large
industries increase. The USA tolerance $p_{c}$ first increases from 1995 to
2000, then slightly decreases from 2000 to 2009, with a small increase in
2010-2011. Germany’s tolerance $p_{c}$ in general slightly increases in the
17-year period and shows small fluctuations between 2000 and 2005.

To further validate these results, we compare the total product output (see
Figure 6, red triangle) and average tolerance $p_{c}$ (see Figure 6, black
circles) for China, USA, and Germany, as a function of time. The product
output (see Figure 6, red triangle) value is the total money flow a country
supplies to the other countries plus value added in the products, which also
indicates its total trade impact on foreign countries. Studying Figure 6, we
find that generally speaking the total outputs of the three investigated
countries grows throughout time, with that of the USA and China being higher
in value than that of Germany (see Figure 6, right y-axis). However, by
comparing the tolerance $p_{c}$, we find three different behaviors. First, for
the case of China (Figure 6(a)), we find that both the tolerance $p_{c}$ and
total outputs are growing in time, and observe that in the early 2000’s there
was a jump in the tolerance $p_{c}$, following by a sharp increase in the
total output. Secondly, for the case of the USA (Figure 6(b)), we find that
while the total output is increasing in time, the tolerance $p_{c}$ is in a
decreasing trend. Finally, for the case of Germany (Figure 6(c)), we find that
while the total import is increasing in time, the tolerance $p_{c}$ is rather
stable with small fluctuations.

Figure 6: Tolerance $p_{c}$ of China, the USA and Germany comparing to the
total product output value. For each country, the $p_{c}$ is an average of the
largest four industry $p_{c}$ of this country (black circles). The product
output (red triangle) value is the money flow a country supplies to the rest
of the countries, which also indicates its impact to foreign countries. In (a)
and (c), the trends of the $p_{c}$ and output value are similar, which
indicate that China and Germany become more influential to the world economy.
In (b), the $p_{c}$ of the USA increases in the period of year 1995 to year
2000, and decreases from 2000 to 2011, while the output value increases from
1995 to 2011 in general.

This provides further evidence into the change in influence of these three
important economies - that of China is increasing, that of the US is
declining, and that of Germany is rather constant across time. Comparing the
total output to the tolerance $p_{c}$ provides further evidence that the
tolerance measurement of country’s impact reveals the underlying economic
evolving dependencies which is not obvious from the simple measurement of
total output capability.

##  VII Conclusions

We developed a framework to quantify interdependencies in the world industrial
network and measure risk levels in global markets. We use the methodology to
rank the economic importance of each industry and country according to the
global damage that would result from their failures. Using network science to
investigate input-output data of money flows between different economic
industries, it becomes possible to stress test the global economic network and
identify vulnerabilities and sources of systemic risk. Our quantitative
results shed new light on China’s increasing economic influence over other
economies, including that of the USA. The resulting information will enable
policy and decision makers to better measure, understand, and maintain
financial stability.

###### Acknowledgements.

We would like to thank the European Commission FET Open Project FOC 255987 and
FOC-INCO 297149 for financial support. We wish to thank the ONR, DTRA, DFG, EU
project, Multiplex and LINC, EU projects, the Keck Foundation, and the Israel
Science Foundation for financial support.

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