Interdependence of Sectors of Economic Activities for World Countries from the Reduced Google Matrix Analysis of WTO Data

Abstract

We apply the recently developed reduced Google matrix algorithm for the analysis of the OECD-WTO World Network of Economic Activities. This approach allows to determine interdependencies and interactions of economy sectors of several countries, including China, Russia and the USA, properly taking into account the influence of all the other world countries and their economic activities. Within this analysis, we also obtain the sensitivity of EU countries’ economies to the petroleum activity sector. We show that this approach takes into account the multiplicity of economical interactions between countries and activity sectors, thus providing a richer analysis compared to the usual export-import analysis.

Full text

entropy
Article
Interdependence of Sectors of Economic Activities for
World Countries from the Reduced Google Matrix
Analysis of WTO Data
Célestin Coquidé 1, José Lages 1,* and Dima L. Shepelyansky 2
1Institut UTINAM, OSU THETA, Université Bourgogne Franche-Comté, CNRS, 25000 Besançon, France;
celestin.coquide@utinam.cnrs.fr
2
Laboratoire de Physique Théorique, IRSAMC, Université de Toulouse, CNRS, UPS, 31062 Toulouse, France;
dima@irsamc.ups-tlse.fr
*Correspondence: jose.lages@utinam.cnrs.fr
Received: 10 November 2020; Accepted: 10 December 2020; Published: 13 December 2020


Abstract:
We apply the recently developed reduced Google matrix algorithm for the analysis
of the OECD-WTO World Network of Economic Activities. This approach allows to determine
interdependencies and interactions of economy sectors of several countries, including China,
Russia and the USA, properly taking into account the influence of all the other world countries
and their economic activities. Within this analysis, we also obtain the sensitivity of EU countries’
economies to the petroleum activity sector. We show that this approach takes into account the
multiplicity of economical interactions between countries and activity sectors, thus providing a richer
analysis compared to the usual export-import analysis.
Keywords: World Trade Organization; networks; Google matrix; Markovian process; PageRank
1. Introduction
The statistical data of UN COMTRADE [
1
] and the World Trade Organization (WTO) Statistical
Review 2018 [
2
] demonstrate all the complexity of the international trade and the economic relations
between world countries. The world economy and the international trade are mutually interacting
which makes the analysis of their development very important but also complicated [
3
]. Thus,
developed advanced mathematical tools are required for the scientific analysis of such complex systems.
Usually, their analysis uses the matrix tools of input-output transactions broadly applied in economy and
initiated by the fundamental works of Leontief [4,5]. More recent developments are described in [6].
The complex networks constitute a domain of research which emerged in the last two decades
alongside with the development of the modern society generating enormous amount of communication
including the World Wide Web (WWW), Wikipedia, Facebook, Twitter (see e.g., [
7
]). The PageRank
algorithm, developed by Brin and Page in 1998 [
8
] to retrieve information from the WWW, was at
the mathematical foundation of the Google search engine (see e.g., [
9
]). This algorithm constructs the
Google matrix
G
describing Markov chain transitions between the nodes of the WWW network and
allows it to rank billions of web pages of the WWW. The efficient applications of the Google matrix
analysis to various directed networks have been demonstrated in [10].
The application of the Google matrix approach to the World Trade Network (WTN) was pushed
forward in [
11
,
12
] using the UN COMTRADE database [
1
] which contain information for almost
50 years of world trade. In addition to the PageRank algorithm, it was shown that the analysis of
the WTN with the CheiRank algorithm [
13
,
14
], assuming inverted links, plays also an important
role for the study of the world trade. Indeed, the PageRank probabilities of nodes are on average
Entropy 2020,22, 1407; doi:10.3390/e22121407 www.mdpi.com/journal/entropy

Entropy 2020,22, 1407 2 of 21
proportional to the number of ingoing links characterizing the import capabilities of the economic
actors while the CheiRank probabilities are on average proportional to the number of outgoing links,
thus, characterizing export capabilities [
11
,
12
,
15
]. Since both export and import have to be taken
into account to correctly describe the world trade, this clearly shows the importance of the combined
PageRank-CheiRank analysis. A peculiar feature of the Google matrix approach is the democratic
treatment of world countries. They are treated independently of their richness which is different from
the usual import and export ranking. The contributions of the various exchanges of products are taken
to be proportional to their trade volume.
While the UN COMTRADE database contains an enormous amount of information for all the UN
countries with thousands of exchanged products, it records gross flows of commodities and services
and consequently counts the value of a product several times as it crosses borders. In the present
study, we use the OECD-WTO Trade in Value Added (TiVA) database, which is an Input-Output
database, containing net flows of products excluding the contributions of intermediate inputs from
upstream industries. The Google matrix analysis of the World Network of Economic Activities (WNEA)
constructed from the OECD-WTO TiVA has been reported in [
16
]. The approach developed in the
OECD-WTO WNEA (2013) incorporates naturally the economic flows between activity sectors [
16
]
which by construction were absent from the UN COMTRADE based WTN [11,12].
Hence, the new important element of the WNEA is the presence of direct interactions between
the economical sectors. It is interesting to know what are the differences and the similarities of the
economic transfers between the sectors of a specific country taking into account their exchanges with
the sectors of the other world countries. To obtain the interactions between sectors of a given country,
one should take into account their direct interactions (direct links) but also all the indirect pathways
of product transfers throughout the multiplicity of exchanges with the rest of the world. The most
appropriate mathematical tool for the extraction of such direct and indirect interactions is the reduced
Google matrix algorithm (REGOMAX) [
17
]. The efficiency of this approach has been demonstrated in
various field such as Wikipedia networks (e.g., interactions between politicians [
18
], ranking of and
interactions between world universities [
19
], interactions between the largest world banks [
20
]) and
biological networks encoding protein-protein interactions [
21
]. Recently, the REGOMAX analysis of the
UN COMTRADE database allowed to obtain the influence of the petroleum and the gas trades on the
economy of the EU countries [
22
]. Here, we use the REGOMAX approach to obtain the interdependence
of economic sectors for world countries from the OECD-WTO TiVA database (WTO data) already
studied in [
16
]. We note that other similar input-output databases exist such as the World Input-Output
Database (WIOD) [
23
]. In principle, the reduced Google matrix analysis can be also applied to these
databases and it would be interesting to probe the similarity of the main conclusions of the present
study with the ones we could obtained with, e.g., WIOD.
Previous investigations of the World Trade Network data sets have been realized (see e.g., [
24
–
29
]),
however the main different feature with our approach is the use of the Google matrix methods which
characterize both import and export flows taking into account the whole transfer chain between the
nodes of the global network. The analysis of both the import and the export directions is rather rare,
see e.g., [
30
] where the hubs and authorities of the WTN have been studied, but we think that the
Google matrix analysis using the PageRank, the CheiRank and the REGOMAX tools characterizes
the economical activities in at a deeper and a more detailed level. The matrix analysis of the financial
risks already demonstrated its efficiency for undirected flows [
31
–
33
]. However, the financial and
trade flows are directional, and thus, we hope that the Google matrix tools used here will find further
useful applications for the study of financial flows and the understanding of economy complexity.
The recent studies of directed interbank interactions [
34
,
35
] indicate possible interesting applications
of the Google matrix analysis to the study of financial flows between banks. Recent studies [
36
–
38
] also
measure the PageRank centrality of production networks and Global Value Chains. Here, we go well
beyond this sole measure: (1) we take into account the full Google matrix description of the problem as
we build two Google matrices. A first one associated to the economical network with the links giving

Entropy 2020,22, 1407 3 of 21
the direction of the flow of goods, and the second one with inverted links. This description allows
us to determine a PageRank-CheiRank balance for economical sectors and/or for countries which
carries more information than the usual accounting import-export balance [
22
] as it takes into account
the complexity of the entanglement of countries and economical sectors. (2) Moreover, we apply the
REGOMAX algorithm allowing to, for example, extract the direct and the effective indirect links of a
given country economical sectors taken into account the complete information embedded in the global
network describing the complex exchanges between all the sectorial activities of all the countries.
We suppose that the REGOMAX algorithm, developed from the physical problems of quantum
scattering [
17
], can become a useful tool for research in the field of econophysics [
39
]. We note that the
concept of entropy characterizes a possible information amount stored in a system [
40
]. In a steady-state
(like in a thermal equilibrium), the system is characterized by a certain thermal like distribution when
all the information flows inside the system are equilibrated. The Google matrix elements describe the
transition probabilities between system’s sites and the information flows between them. The stationary
probabilities over the sites are given by the PageRank and the CheiRank vectors. Thus, the Google
matrix analysis provides an extension of entropy-type description to network systems.
2. Methods and Data Description
2.1. WNEA Data Sets
As in [
16
], we use the data available from the OECD-WTO TiVA database released in May
2013 which covers the years 1995, 2000, 2005, 2008, 2009. The network contains
Nc=
58 world
countries (57 plus 1 for the Rest Of the World ROW) given in Table 1 in [
16
]. It contains the main
world countries. We do not reproduce this list here since we concentrate our analysis only on several
leading countries with the main emphasis on USA, Russia and China. We use for countries ISO-3166-1
alpha-3 code available at Wikipedia. There are also
Ns=
37 sectors of economic activities given in
Table 1. The sectors are classified according to the International Standard Industrial Classification of
All Economic Activities (ISIC) Rev.3 described in [
1
] and in Wikipedia. We take into account all the
37 sectors, noting that the sectors
s=
1,
. . .
, 21 represent production activities while
s=
22,
. . .
, 37
represent service activities. We concentrate our analysis on sectors
s=
1,
. . .
, 21. The total size of
the Google matrix is
N=NcNs=
58
×
37
=
2146. The main analysis is presented for year 2008.
Additional data for other years are available upon request. In addition, all the OECD-WTO TiVA
network data are available upon request [41].
Table 1.
List of sectors considered by input/output matrices from the WTO-OECD database,
their correspondence to the ISIC UN classification is also given. The second column gives the short
names we use in the present paper to designate the different economical sectors.
sShort Name OECD ICIO Category ISIC Rev. 3 Correspondence
1agriculture C01T05 AGR
01—Agriculture, hunting and related service activities
02—Forestry,logging and related service activities
05—Fishing, operation of fish hatcheries and fish farms; service activities incidental to fishing
2mining C10T14 MIN
10—Mining of coal and lignite; extraction of peat
11—Extraction of crude petroleum and natural gas; service activities incidental to oil and gas extraction excluding surveying
12—Mining of uranium and thorium ores
13—Mining of metal ores
14—Other mining and quarrying
3food C15T16 FOD 15—Manufacture of food products and beverages
16—Manufacture of tobacco products
4textile C17T19 TEX
17—Manufacture of textiles
18—Manufacture of wearing apparel; dressing and dyeing of fur
19—Tanning and dressing of leather; manufactureof luggage, handbags, saddlery, harness and footwear
5wood C20 WOD 20—Manufacture of wood and of products of wood and cork, except furniture;
Manufacture of articles of straw and plaiting materials
6paper C21T22 PAP 21—Manufacture of paper and paper products
22—Publishing, printing and reproduction of recorded media
7petroleum C23 PET 23—Manufacture of coke, refined petroleum products and nuclear fuel
8chemicals C24 CHM 24—Manufacture of chemicals and chemical products

Entropy 2020,22, 1407 4 of 21
Table 1. Cont.
sShort Name OECD ICIO Category ISIC Rev. 3 Correspondence
9plastics C25 RBP 25—Manufacture of rubber and plastics products
10 minerals C26 NMM 26—Manufacture of other non-metallic mineral products
11 metals C27 MET 27—Manufacture of basic metals
12 metal prod. C28 FBM 28—Manufacture of fabricated metal products, except machinery and equipment
13 equipment C29 MEQ 29—Manufacture of machinery and equipment n.e.c.
14 office mach. C30 ITQ 30—Manufacture of office, accounting and computing machinery
15 electrical mach. C31 ELQ 31—Manufacture of electrical machinery and apparatus n.e.c.
16 com. equipment C32 CMQ 32—Manufacture of radio, television and communication equipment and apparatus
17 medical C33 SCQ 33—Manufacture of medical, precision and optical instruments, watches and clocks
18 motor C34 MTR 34—Manufacture of motor vehicles, trailers and semi-trailers
19 trans. equip. C35 TRQ 35—Manufacture of other transport equipment
20 furniture C36T37 OTM 36—Manufacture of furniture; manufacturing n.e.c.
37—Recycling
21 elec/gas/water C40T41 EGW 40—Electricity, gas, steam and hot water supply
41—Collection, purification and distribution of water
22 construction C45 CON 45—Construction
23 vehicles C50T52 WRT
50—Sale, maintenance and repair of motor vehicles and motorcycles; retail sale of automotive fuel
51—Wholesale trade and commission trade, except of motor vehicles and motorcycles
52—Retail trade, except of motor vehicles and motorcycles; repair of personal and household goods
24 hotels/restaurants C55 HTR 55—Hotels and restaurants
25 transport C60T63 TRN
60—Land transport; transport via pipelines
61—Water transport
62—Air transport
63—Supporting and auxiliary transport activities; activities of travel agencies
26 telecom C64 PTL 64—Post and telecommunications
27 financial C65T67 FIN
65—Financial intermediation, except insurance and pension funding
66—Insurance and pension funding, except compulsory social security
67—Activities auxiliary to financial intermediation
28 real estate C70 REA 70—Real estate activities
29 renting equip. C71 RMQ 71—Renting of machinery and equipment without operator and of personal and household goods
30 computer C72 ITS 72—Computer and related activities
31 R&D C73 RDS 73—Research and development
32 other1 C74 BZS 74—Other business activities
33 public C75 GOV 75—Public administration and defense; compulsory social security
34 education C80 EDU 80—Education
35 health C85 HTH 85—Health and social work
36 other2 C90T93 OTS
90—Sewage and refuse disposal, sanitation and similar activities
91—Activities of membership organizations n.e.c.
92—Recreational, cultural and sporting activities
93—Other service activities
37 private C95 PVH 95—Private households with employed persons
2.2. Google Matrix Construction for WNEA
In the following, we use the approach developed in [
12
,
16
] to construct the Google matrix of the
economical transfers between the activity sectors of the different countries. We keep the notations used
in [16].
From the WTO data, we construct the matrix
Mcc0,ss0
of money transfer between nodes expressed
in USD of the current year
Mcc0,ss0=transfer from country c0, sector s0to country c, sector s. (1)
Here the country indexes are
c
,
c0∈ {
1,
. . .
,
Nc}
and the activity sector indexes are
s
,
s0∈ {
1,
. . .
,
Ns}
with
Nc=
58 and
Ns=
37. Here, each node represents a pair of country and activity sector. A link
gives the transfer a sector of one country to another sector of another country. We construct the matrix
Mcc0,ss0
from the TiVA Input/Output tables using the transposed representation so that the volume of
the products or the sectors flows in a column from line to line; for a given country
c
we exclude possible
exchanges
(c
,
s)→(c
,
s)
from a sector
s
to itself. The matrix construction of
Mcc0,ss0
highlights the trade
exchange flows intra- and inter-countries.

Entropy 2020,22, 1407 5 of 21
The Google matrices Gand G∗are N×Nmatrices with real non-negative elements defined as
Gij =αSij + (1−α)vi, and, G∗i j =αS∗ij + (1−α)v∗
i, (2)
where
N=Nc×Ns
,
α
is the damping factor (0
<α<
1), and
v
is a positive column vector called
personalization vector with the normalization
∑ivi=
1 [
9
,
12
]. We note that the usual Google matrix
corresponds to a personalization vector with
vi=
1
/N
. Here as in [
11
,
12
], we fix
α=
0.5 noting
that a variation of
α
in a range 0.5–0.9 does not significantly affect the probability distributions of
the PageRank and the CheiRank vectors [
9
–
11
]. The personalization vector is taken from the vector
representing the exchange weight of each sector as it is described in [
16
] (for the multiproduct WTN
the same choice of this vector is described in [
12
,
22
]). As in [
12
,
16
], we call this approach the Google
Personalized Vector Method (GPVM).
The matrices Sand S∗are built from money matrices Mcc0,ss0as
Si,i0=(Mcc0,ss0/V∗
c0s0if V∗
c0s06=0
1/Nif V∗
c0s0=0
S∗
i,i0=(Mc0c,s0s/Vc0s0if Vc0s06=0
1/Nif Vc0s0=0(3)
where
i(0)=s(0)+ (c(0)−
1
)Ns∈ {
1,
. . .
,
N}
. We have also defined
Vcs =∑c0s0Mcc0,ss0
and
V∗
cs =
∑c0s0Mc0c,s0s
which are the total volume of import and export for the sector
s
of country
c
. The sum
of the elements of each column of
S
and
S∗
is normalized to unity and
G
,
G∗
,
S
, and
S∗
belong to the
class of Google matrices. The import properties are characterized by
S
and
G
, and export properties by
S∗
and
G∗
. Let us note that the starred matrices,
G∗
and
S∗
, are built in the same manner as the other
matrices,
G
and
S
, but from the network for which all the directions of the links have been inverted.
Consequently, the starred matrices,
G∗
and
S∗
are build from the transpose of the money matrix
M
(1).
The PageRank and CheiRank vectors are right eigenvectors of the matrices
G
and
G∗
with the
eigenvalue
λ=
1. Their components are positive nonzero real numbers and their sum is normalized
to unity. The components give the probabilities to find a random seller (surfer) on a given node after
a long walk over the network. The PageRank index
K
and the CheiRank index
K∗
are defined by
the components of the PageRank vector
P
and the CheiRank vector
P∗
sorted by descending order,
P(K)≥P(K+
1
)
and
P∗(K∗)≥P∗(K∗+
1
)
with
K
,
K∗=
1,
. . .
,
N
. Since we have countries
c
and
economic sectors
s
, it is convenient to use two indexes probabilities
Pcs
and
P∗
cs
with 1
≤c≤
58 and
1≤s≤37. The sum over all the sectors gives the probabilities Pcand P∗
cfor each country c.
2.3. Reduced Google Matrix for WNEA
The REGOMAX algorithm, proposed in [
17
], is described in detail in [
18
]. Here we give the main
elements of this method keeping the notations of [18,22].
The reduced Google matrix
GR
is constructed for a selected subset of
Nr
nodes. The construction
is based on concepts of scattering theory used in different fields including mesoscopic physics,
nuclear physics, and quantum chaos. It captures, in a matrix of size
Nr×Nr
, the full contribution of
direct and indirect pathways existing in the global network of
N
nodes between the
Nr
selected nodes
of interest. The PageRank probabilities of the
Nr
nodes are the same as for the global network with
N
nodes up to a global constant factor taking into account that the sum of PageRank probabilities over
Nr
nodes is unity. The
(i
,
j)
-element of
GR
can be interpreted as the probability for a random seller
(surfer) starting at node
j
to arrive in node
i
using direct and indirect interactions. Indirect interactions
refer to pathways composed in part with nodes different from the
Nr
ones of interest. The computation
steps of
GR
offer a decomposition of
GR
into matrices that clearly distinguish direct from indirect
interactions: GR=Grr +Gpr +Gqr [18]. Here, the Grr matrix is generated by the direct links between
selected
Nr
nodes in the global
G
matrix with
N
nodes. The
Gpr
matrix is usually rather close to a

Entropy 2020,22, 1407 6 of 21
matrix for which each column reproduces the PageRank vector
Pr
associated to the
Nr
nodes of interest.
Due to that,
Gpr
does not bring much information about direct and indirect links between selected
nodes. The interesting role is played by
Gqr
. It takes into account all the indirect links between the
Nr
selected nodes appearing due to the myriads of pathways passing via the rest of the
N−Nr'N
nodes of the global network (see [
17
,
18
]). The matrix
Gqr =Gqrd +Gqrnd
has a diagonal part (
Gqrd
)
and a non-diagonal part (
Gqrnd
). The
Gqrnd
matrix describes indirect interactions between different
nodes. The explicit formulas of the mathematical and numerical computation methods of all three
matrix components of GRare given in [17,18,22].
2.4. Sensitivity of the Economy Balance
As in [
22
], within the REGOMAX approach, we determine the whole economy balance of a given
country with PageRank and CheiRank probabilities as
Bc= (P∗
c−Pc)/(P∗
c+Pc)
. The sensitivity
of the country
c
economy balance
Bc
to the price of, e.g., the sector
s
of petroleum can be obtained:
(1) by changing the corresponding money volume flow related to this sector multiplying it by
(
1
+δ)
,
(2) by recomputing all the rank probabilities, and then 3- by computing the derivative
D(s→c) =
dBc/dδ. This approach was explained and used in [12,22].
We can also use the same procedure to determine, for a given country, the sensitivity of its economy
sector balance
Bcs = (P∗
cs −Pcs)/(P∗
cs +Pcs)
to the price variation of the petroleum sector
s
. Then the
sensitivity of a sector s0of a given country cto another sector sis defined as D(s→cs0) = dBcs0/dδ.
The usual export-import description of economic activities (and also trade exchange) between
countries takes into account only direct seller-buyer interactions. In contrast, the Google matrix
approach takes into account all the chains of transactions between countries which in a steady-state
limit are characterized by the PageRank and CheiRank vectors. It has been already shown that
compared to the simple import-export description the Google matrix analysis gives a complementary
more detailed and deep description of interactions between countries and sectors of economic
activity [
12
,
16
,
22
]. This provides an additional relevant characterization of these interactions for policy
makers allowing a better understanding of non obvious indirect economic ties between countries.
As an example, our approach can be used by policy makers as a tool to contain economical crisis
contagion [15].
3. Results
Hereafter, the economical sectors are designated by the short names given in the second column
of the Table 1. For the sake of clarity, these sector short names are printed in boldface.
3.1. Interdependence of the USA Economy Sectors
The reduced Google matrices
GR
and
G∗
R
and their three matrix components for PageRank and
CheiRank algorithms with
Nr=
21 sectors of USA economy activity (
s=
1, ..., 21 in Table 1) are
shown in Figures 1and 2respectively. As shown in [
18
,
22
], it is useful to characterize each matrix
component by their weights
Wpr
,
Wrr
,
Wqr
(and
Wqrnd
) corresponding to
Gpr
,
Grr
,
Gqr
(and
Gqrnd
).
For each component, the weight is defined as the sum of all the matrix elements divided by the matrix
size, i.e.,
Nr
. By definition
Wpr +Wrr +Wqr =
1. For the USA, the matrix weights are given in the
captions of Figures 1and 2. For Wikipedia networks (see, e.g., [
18
–
20
]), one usually has the weights
Wpr ≈
0.95 and
Wrr ≈Wqr ≈
0.025 [
18
]. Here the situation is more similar to the WTN case [
22
] where
the weight of
Wqr
remains rather small while
Wrr
is by a factor of 3–10 larger than in Wikipedia. As for
the WTN [
22
], we attribute this to a significantly larger number of links per node in the WTN and
WNEA global networks in comparison with Wikipedia networks.
The strongest matrix elements in
GR
show the interdependence of sectors of USA for import or
PageRank direction (see Figure 1). Here, we see the dominance of the interaction from the
agriculture
sector (
s=
1) to the
food
sector (
s=
3). Indeed, the agricultural activity produces food used by all
the people that makes this link so strong. Another strong link is from
mining
(
s=
2) to
petroleum

Entropy 2020,22, 1407 7 of 21
(
s=
7). Indeed, coke and petroleum are produced by mining and they play an important role in the
USA economy. The third link by strength is from
metals
(
s=
11, manufacture of basic metals) to
metal
prod.
(
s=
12, manufacture of fabricated metal products) that is also very natural. These three links
are also well present among direct links in
Grr
that corresponds to the importance of direct links in the
WNEA discussed above. Interestingly, other links, from
paper
(paper and paper product) and from
plastics
(rubber and plastics products) to
food
, which are not so strong in the direct matrix component
Grr
are enhanced in
GR
, illustrating the role of indirect interactions. Indeed, the food products industry
uses products of paper and plastic industries for packaging.
agriculture
mining
food
textile
wood
paper
petroleum
chemicals
plastics
minerals
metals
metal prod.
equipment
office mach.
electrical mach.
com. equipment
medical
motor
trans. equip.
furniture
elec/gas/water
agriculture
mining
food
textile
wood
paper
petroleum
chemicals
plastics
minerals
metals
metal prod.
equipment
office mach.
electrical mach.
com. equipment
medical
motor
trans. equip.
furniture
elec/gas/water
agriculture
mining
food
textile
wood
paper
petroleum
chemicals
plastics
minerals
metals
metal prod.
equipment
office mach.
electrical mach.
com. equipment
medical
motor
trans. equip.
furniture
elec/gas/water
agriculture
mining
food
textile
wood
paper
petroleum
chemicals
plastics
minerals
metals
metal prod.
equipment
office mach.
electrical mach.
com. equipment
medical
motor
trans. equip.
furniture
elec/gas/water
GRGpr
Grr Gqr
Figure 1.
Density plot of reduced Google matrix for import or PageRank direction:
GR
(
top left
),
Gpr
(
top right
),
Grr
(
bottom left
) and
Gqr
without diagonal elements (
bottom right
). The matrices
are computed for a set of reduced nodes composed of
Nr=
21 sectors (
s=
1,
. . .
, 21) of USA for the
year 2008. The corresponding matrix weights are:
Wpr =
0.813817,
Wrr =
0.155258,
Wqr =
0.030925
and
Wqrnd =
0.027383. For each panel, each cell corresponds to a given value of the Google matrix
component (
GR
,
Gpr
,
Grr
, or
Gqr
), the colorbar gives the correspondence between matrix elements
values and colors (from blue for 0 to red for the maximum). This value characterizes the intensity of
the interaction between two economical sectors. The direction of the interaction is from bottom to left.
Among the
Gpr
matrix components, there are three dominant horizontal lines for
food
,
motor
(motor vehicles), and
chemicals
, which are pointed by the majority of the sectors. These three sectors,
which are major sector activities using products of many others, are at the top 3 PageRank positions of
USA sectors.
The
Gqr
matrix components highlight hidden links between USA economic sectors. Among the
most pronounced hidden interactions in
Gqr
we note that:
food
is pointing to
paper
,
petroleum
is

Entropy 2020,22, 1407 8 of 21
pointing to
food
and
paper
. It is clearly understandable that food and paper industries indirectly use
petroleum products. Concerning the
food
to
paper
link, according to
Grr
, the food industry directly
points to agriculture, and of course the paper industry uses silviculture. This is one of the many
possible indirect paths linking food to paper.
For the export or CheiRank direction, the results are shown in Figure 2. Here, the strongest
links are from
plastics
(
s=
9, manufacture of rubber and plastics products) to
chemicals
(
s=
8,
manufacture of chemicals), from
textile
(
s=
4, manufacture of textiles) to
chemicals
(
s=
8), and from
petroleum
to
mining
. Here again, the dominant contribution is given by
G∗
rr
but the strength of the
final amplitudes is slightly corrected by
G∗
pr
and
G∗
qr
contributions which mainly highlight the fact that
the petroleum and chemistry industries are the main suppliers of the other economic activity sectors.
0
0.05
0.1
0.15
0.2
0.25
0.3
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
GRGpr
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
Grr Gqr
agriculture
mining
food
textile
wood
paper
petroleum
chemicals
plastics
minerals
metals
metal prod.
equipment
office mach.
electrical mach.
com. equipment
medical
motor
trans. equip.
furniture
elec/gas/water
agriculture
mining
food
textile
wood
paper
petroleum
chemicals
plastics
minerals
metals
metal prod.
equipment
office mach.
electrical mach.
com. equipment
medical
motor
trans. equip.
furniture
elec/gas/water
agriculture
mining
food
textile
wood
paper
petroleum
chemicals
plastics
minerals
metals
metal prod.
equipment
office mach.
electrical mach.
com. equipment
medical
motor
trans. equip.
furniture
elec/gas/water
agriculture
mining
food
textile
wood
paper
petroleum
chemicals
plastics
minerals
metals
metal prod.
equipment
office mach.
electrical mach.
com. equipment
medical
motor
trans. equip.
furniture
elec/gas/water
Figure 2.
Density plot of reduced Google matrix for export or CheiRank direction:
G∗
R
(
top left
),
G∗
pr
(
top right
),
G∗
rr
(
bottom left
) and
G∗
qr
without diagonal elements (
bottom right
). The matrices
are computed for a set of reduced nodes composed of
Nr=
21 sectors (
s=
1,
. . .
, 21) of the USA for
the year 2008. The corresponding matrix weights are:
W∗
pr =
0.78968,
W∗
rr =
0.18289,
W∗
qr =
0.02743
and
W∗
qrnd =
0.02554. For each panel, each cell corresponds to a given value of the Google matrix
component (
G∗
R
,
G∗
pr
,
G∗
rr
, or
G∗
qr
), the colorbar gives the correspondence between matrix elements
values and colors (from blue for 0 to red for the maximum). This value characterizes the intensity of
the interaction between two economical sectors. The direction of the interaction is from bottom to left.
The amplitudes of all matrix elements of
GR
for the USA, Russia (RUS), and China (CHN) and for
the different years are available upon request.

Entropy 2020,22, 1407 9 of 21
3.2. Interdependence of the Russian Economy Sectors
The reduced Google matrices for Russia for PageRank (import) and CheiRank (export) directions
are shown in Figures 3and 4respectively. They are constructed in the same manner as Figures 1and 2
for USA.
For the reduced Google matrix
GR
of Russian economic sectors with PageRank (import) direction,
see Figure 3, the strongest link is between
agriculture
and
food
. This is similar to previous depicted
case of USA. We note here that the inverted link, i.e., from
food
to
agriculture
, is weaker but also
present in
GR
and in
Grr
. This is certainly due to the fact that the food industry also produces products
for animal used in agriculture. In
Gpr
, the most pronounced horizontal line is for
food
, highlighting the
fact that this industry uses indirectly products of almost all the other economic sectors; it is followed
by the line of
motor
(
s=
18, manufacture of motor vehicles) and
elec/gas/water
(
s=
21). Among
indirect links, the strongest one is from
minerals
(
s=
10, manufacture of other non-metallic mineral
products) to elec/gas/water.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0
0.05
0.1
0.15
0.2
0.25
0
0.005
0.01
0.015
0.02
0.025
GRGpr
Grr Gqr
agriculture
mining
food
textile
wood
paper
petroleum
chemicals
plastics
minerals
metals
metal prod.
equipment
office mach.
electrical mach.
com. equipment
medical
motor
trans. equip.
furniture
elec/gas/water
agriculture
mining
food
textile
wood
paper
petroleum
chemicals
plastics
minerals
metals
metal prod.
equipment
office mach.
electrical mach.
com. equipment
medical
motor
trans. equip.
furniture
elec/gas/water
agriculture
mining
food
textile
wood
paper
petroleum
chemicals
plastics
minerals
metals
metal prod.
equipment
office mach.
electrical mach.
com. equipment
medical
motor
trans. equip.
furniture
elec/gas/water
agriculture
mining
food
textile
wood
paper
petroleum
chemicals
plastics
minerals
metals
metal prod.
equipment
office mach.
electrical mach.
com. equipment
medical
motor
trans. equip.
furniture
elec/gas/water
Figure 3.
The same as in Figure 1for Russia (RUS) in 2008. The corresponding matrix weights are:
Wpr =
0.851677,
Wrr =
0.112809,
Wqr =
0.035514 and
Wqrnd =
0.033682. For each panel, each cell
corresponds to a given value of the Google matrix component (
GR
,
Gpr
,
Grr
, or
Gqr
), the colorbar
gives the correspondence between matrix elements values and colors (from blue for 0 to red for the
maximum). This value characterizes the intensity of the interaction between two economical sectors.
The direction of the interaction is from bottom to left.
For the reduced Google matrix
G∗
R
of the Russian economic sectors with CheiRank (export)
direction, see Figure 4, there is the dominance of lines related to
mining
, followed, with weaker

Entropy 2020,22, 1407 10 of 21
intensities, by
petroleum
,
metals
, and
elec/gas/water
. This picture is rather different from the USA
case in Figure 2. Although there are only very few weak direct links pointing to
mining
, the mining
sector is very important since, through the network of exports, it strongly acts (via almost only
indirect interactions) upon every sector of the Russian economy. For the hidden links encoded in
G∗
qr
,
the dominant line is for elec/gas/water.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0
0.005
0.01
0.015
0.02
0.025
GRGpr
Grr Gqr
agriculture
mining
food
textile
wood
paper
petroleum
chemicals
plastics
minerals
metals
metal prod.
equipment
office mach.
electrical mach.
com. equipment
medical
motor
trans. equip.
furniture
elec/gas/water
agriculture
mining
food
textile
wood
paper
petroleum
chemicals
plastics
minerals
metals
metal prod.
equipment
office mach.
electrical mach.
com. equipment
medical
motor
trans. equip.
furniture
elec/gas/water
agriculture
mining
food
textile
wood
paper
petroleum
chemicals
plastics
minerals
metals
metal prod.
equipment
office mach.
electrical mach.
com. equipment
medical
motor
trans. equip.
furniture
elec/gas/water
agriculture
mining
food
textile
wood
paper
petroleum
chemicals
plastics
minerals
metals
metal prod.
equipment
office mach.
electrical mach.
com. equipment
medical
motor
trans. equip.
furniture
elec/gas/water
Figure 4.
The same as in Figure 2for Russia (RUS) in 2008. The corresponding matrix weights are:
W∗
pr =
0.804255,
W∗
rr =
0.159634,
W∗
qr =
0.036111 and
W∗
qrnd =
0.033377. For each panel, each cell
corresponds to a given value of the Google matrix component (
G∗
R
,
G∗
pr
,
G∗
rr
, or
G∗
qr
), the colorbar
gives the correspondence between matrix elements values and colors (from blue for 0 to red for the
maximum). This value characterizes the intensity of the interaction between two economical sectors.
The direction of the interaction is from bottom to left.
3.3. Interdependence of the Chinese Economy Sectors
Interdependencies of the economy sectors of China for PageRank (import) and CheiRank (export)
directions are presented with the reduced Google matrix and its components in
Figures 5and 6
respectively.
For the reduced Google matrix
GR
, shown in Figures 5, there are strong links between
agriculture
and
food
similarly to the USA and Russian cases. In addition, there is a strong transition from
com.
equipment
(
s=
16, manufacture of communication equipment) to
office mach.
(
s=
14, manufacture
of computing machinery). This corresponds to strong Chinese production of TV, computers and
other communication related products. In the matrix component
Gpr
, there are strong lines for
food
,
chemicals
,
equipment
(machinery and equipment), and
com. equipment
. We note that, contrary to
the USA and Russia, the food sector does not dominate alone as top importer. Indeed, chemistry,

Entropy 2020,22, 1407 11 of 21
communication, computing, and machinery industries also play important roles as they also indirectly
use products of many Chinese economic sectors. For the hidden links in the matrix component
Gqr
,
the strongest matrix element points from
com. equipment
to
office mach.
. Many other links, with a
slightly weaker intensity, are also highlighted by the
Gqr
matrix component, but these are quite weak
in comparison with the highest intensities in the reduced Google matrix GR.
0
0.05
0.1
0.15
0.2
0.25
0.3
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0
0.05
0.1
0.15
0.2
0.25
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
GRGpr
Grr Gqr
agriculture
mining
food
textile
wood
paper
petroleum
chemicals
plastics
minerals
metals
metal prod.
equipment
office mach.
electrical mach.
com. equipment
medical
motor
trans. equip.
furniture
elec/gas/water
agriculture
mining
food
textile
wood
paper
petroleum
chemicals
plastics
minerals
metals
metal prod.
equipment
office mach.
electrical mach.
com. equipment
medical
motor
trans. equip.
furniture
elec/gas/water
agriculture
mining
food
textile
wood
paper
petroleum
chemicals
plastics
minerals
metals
metal prod.
equipment
office mach.
electrical mach.
com. equipment
medical
motor
trans. equip.
furniture
elec/gas/water
agriculture
mining
food
textile
wood
paper
petroleum
chemicals
plastics
minerals
metals
metal prod.
equipment
office mach.
electrical mach.
com. equipment
medical
motor
trans. equip.
furniture
elec/gas/water
Figure 5.
Same as in Figure 1for China CHN in 2008. The corresponding matrix weights are:
Wpr =
0.698164,
Wrr =
0.263683,
Wqr =
0.038153 and
Wqrnd =
0.035547. For each panel, each cell
corresponds to a given value of the Google matrix component (
GR
,
Gpr
,
Grr
, or
Gqr
), the colorbar
gives the correspondence between matrix elements values and colors (from blue for 0 to red for the
maximum). This value characterizes the intensity of the interaction between two economical sectors.
The direction of the interaction is from bottom to left.
For the reduced Google matrix
G∗
R
, shown in Figure 6, the strongest matrix elements are from
food
to agriculture, from
metal prod.
to
metals
, from
petroleum
to
mining
, from
plastics
to
chemicals
,
and from
electrical mach.
(electrical machinery and apparatus) to
metals
. For the
G∗
pr
matrix,
the strongest horizontal line of transitions is for
metals
. Among the indirect matrix elements of
G∗
qr, the strongest link is from office mach. to com. equipment.

Entropy 2020,22, 1407 12 of 21
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
GRGpr
Grr Gqr
agriculture
mining
food
textile
wood
paper
petroleum
chemicals
plastics
minerals
metals
metal prod.
equipment
office mach.
electrical mach.
com. equipment
medical
motor
trans. equip.
furniture
elec/gas/water
agriculture
mining
food
textile
wood
paper
petroleum
chemicals
plastics
minerals
metals
metal prod.
equipment
office mach.
electrical mach.
com. equipment
medical
motor
trans. equip.
furniture
elec/gas/water
agriculture
mining
food
textile
wood
paper
petroleum
chemicals
plastics
minerals
metals
metal prod.
equipment
office mach.
electrical mach.
com. equipment
medical
motor
trans. equip.
furniture
elec/gas/water
agriculture
mining
food
textile
wood
paper
petroleum
chemicals
plastics
minerals
metals
metal prod.
equipment
office mach.
electrical mach.
com. equipment
medical
motor
trans. equip.
furniture
elec/gas/water
Figure 6.
The same as in Figure 2for China (CHN) in 2008. The corresponding matrix weights are:
W∗
pr =
0.647087,
W∗
rr =
0.326402,
W∗
qr =
0.026511 and
W∗
qrnd =
0.024648. For each panel, each cell
corresponds to a given value of the Google matrix component (
G∗
R
,
G∗
pr
,
G∗
rr
, or
G∗
qr
), the colorbar
gives the correspondence between matrix elements values and colors (from blue for 0 to red for the
maximum). This value characterizes the intensity of the interaction between two economical sectors.
The direction of the interaction is from bottom to left.
3.4. Intersensitivity of the Economy Sectors
The above results show specific dependencies between economy sectors for USA, Russia,
and China. Here we choose 10 countries (USA, RUS, CHN, DEU, FRA, ITA, GBR, JAP, KOR, IND),
for which we determine the balance sensitivity of each economical activity sectors to a price variation
of a specific sector. Excepting KOR, these countries are among the top 10 importers according to
PageRank algorithm applied to the global Google matrix G. KOR is ranked at the 12th position.
For a given country, the balance sensitivity of a sector
s0
to an infinitesimal increase of sector
s
product prices is
D(cs →cs0) = dBcs0/dδ
. The balance of the economic sector
s
is defined as
Bcs =(P∗
cs −Pcs)/(P∗
cs +Pcs), see Section 2.4 for details.
In Figure 7, we show a map of the balance sensitivity to the
petroleum
sector for years 1995 and
2008. The maximal absolute value of the balance sensitivity
D
is increased approximately by a factor 3
from 1995 to 2008 showing an increased dependence of economy sectors to petroleum. Partially, this
can be attributed to the petroleum price growth from 1995 to 2008 (changed by a factor
∼
3, even
∼
5
taking April 2008 as reference). For both years and for any of the considered countries, we observe
that some economic sectors, such as
electrical mach.
(electrical machinery and apparatus),
medical
(medical, precision and optical instruments, watches and clocks),
trans. equip.
(transport equipment),

Entropy 2020,22, 1407 13 of 21
and
wood
, are almost insensitive to the petroleum products sector. Inversely, the most sensitive
economic sectors to
petroleum
sector are
chemicals
,
metals
,
elec/gas/water
,
food
, and (mostly in
1995)
agriculture
. Indeed, the activities of these industries directly use petroleum products. For each
country, the
mining
sector is robust from 1995 to 2008 except for the Russian mining sector for which
the balance sensitivity goes from
−
0.0045 in 1995 to
−
0.012 in 2008. This is a peculiarity of the Russian
mining sector which appears strongly dependent on the Russian petroleum sector. The same data
as in Figure 7are represented in Figure A1 (see Appendix A) but with the same color scale for 1995
and 2008. In Figure A1, we globally observe that from 1995 to 2008 the chemicals sector increased its
sensitivity to
petroleum
sector, by, e.g., a factor
∼
3 for USA and KOR. A weaker increase of sensitivity
to
petroleum
sector could be observed for the
metals
and
food
sectors. We also observe that Russian
and Japanese
elec/gas/water
sectors become more sensitive to their national
petroleum
sector from
1995 to 2008. From Figure A1, we additionally note that all the economic sectors of Germany, Italy, and,
to a somewhat less extent, of France, Great-Britain, and China remain insensitive to their
petroleum
economic sector from 1995 to 2008.
0
-0.004
-0.003
-0.002
-0.001
-0.01
-0.008
-0.006
-0.004
-0.002
0
agriculture
mining
food
textile
wood
paper
petroleum
chemicals
plastics
minerals
metals
metal prod.
equipment
office mach.
electrical mach.
com. equipment
medical
motor
trans. equip.
furniture
elec/gas/water
agriculture
mining
food
textile
wood
paper
petroleum
chemicals
plastics
minerals
metals
metal prod.
equipment
office mach.
electrical mach.
com. equipment
medical
motor
trans. equip.
furniture
elec/gas/water
Figure 7.
Sector balance sensitivity to
petroleum
sector for year 1995 (
left
) and 2008 (
right
);
horizontal axis represents the sector index in data order; vertical axis represents the country index
in PageRank order for the given year. For each couple
(s
,
c)
we modify the link from (
petroleum
,
c
)
towards
(s
,
c)
and compute the
(s
,
c)
balance sensitivity,
D(cpetroleum
,
cs)
. For each plot, a given
color corresponds to a given intensity of the sensitivity
D(cpetroleum
,
cs)
. Grey column represents
self sensitivity (not shown).
The 2008 sector balance sensitivities to
chemicals
,
metals
,
motor
, and
mining
sectors are shown
in Figure 8. Among these economic sectors, the
chemicals
sector has the broadest impact on the
other economic sectors. The strongest sensitivities to the
chemicals
and the
metals
sectors concern
the Russian
mining
and the German
motor
sectors. The German economy is the most affected by
the
motor
sector, particularly the
equipment
, the
food
, and the
chemicals
sectors. The most sensitive
economic sectors to the
mining
sector are the
petroleum
and the
metals
sectors, particularly the
Russian petroleum and metals sectors and the US petroleum sector.

Entropy 2020,22, 1407 14 of 21
-0.007
-0.006
-0.005
-0.004
-0.003
-0.002
-0.001
0
-0.012
-0.01
-0.008
-0.006
-0.004
-0.002
0
-0.012
-0.01
-0.008
-0.006
-0.004
-0.002
0
-0.01
-0.009
-0.008
-0.007
-0.006
-0.005
-0.004
-0.003
-0.002
-0.001
0
agriculture
mining
food
textile
wood
paper
petroleum
chemicals
plastics
minerals
metals
metal prod.
equipment
office mach.
electrical mach.
com. equipment
medical
motor
trans. equip.
furniture
elec/gas/water
agriculture
mining
food
textile
wood
paper
petroleum
chemicals
plastics
minerals
metals
metal prod.
equipment
office mach.
electrical mach.
com. equipment
medical
motor
trans. equip.
furniture
elec/gas/water
Figure 8.
Sector balance sensitivity to
chemicals
(
top left
),
metals
(
top right
),
motor
(
bottom left
),
and
mining
(
bottom right
) sectors for year 2008; horizontal axis represents the sector index in data
order; vertical axis represents the country index in PageRank order for the given year. For each
couple
(s
,
c)
we modify the link from
(s0
,
c)
towards
(s
,
c)
and compute the
(s
,
c)
balance sensitivity,
D(cs0
,
cs)
. For each panel, a given color corresponds to a given intensity of the represented sensitivity.
Grey column represents self sensitivity (not shown).
3.5. Reduced Network of Economic Sectors
We construct the reduced networks of the 21 economic sectors for different countries. For that
purpose, we use the import reduced Google matrices
GR
and export reduced Google matrix
G∗
R
corresponding to the USA, Russia, and China economic sectors for 2019. Examples of such reduced
Google matrices are presented in Sections 3.1–3.3 for 2008.
For a given country
c
and for the economic sector
s
, we select the four links
{(c
,
s)→
(c
,
si)}i=α,β,γ,δ
giving the strongest entries in the composite matrix
Grr +Gqrnd
(or
G∗
rr +G∗
qrnd
) extracted
from the reduced Google matrix
GR
(or
G∗
R
) associated to the 21 economic sectors
{(c
,
si)}i=1,...,21
(see
Figures 1–6to have an idea of the composite matrices,
Grr +Gqrnd
and
G∗
rr +G∗
qrnd
, for USA, Russia,
and China). The reduced networks constructed from the components of the reduced Google matrix
GR
(
G∗
R
) highlight import (export) capabilities of the economic sectors. Let us note that the compact
picture given by the reduced Google matrices at the level of a country comprises in fact the global
information encoded in the global Google matrix of all the transactions from any sector
s
of a country
c
to any sector s0of a country c0.
In Figure 9(top row) we present the reduced network of US economic sectors for import (left panel)
and export (right panel) exchanges. From the import point of view, we observe that the
chemicals
sector uses products from the broadest variety of US economic sectors as it has the maximum of
ingoing links (13 out of 21 economic sector are pointing to the
chemicals
sector). Other economic
sectors using many US resources are
food
(10 out of 21),
paper
(10 out of 21),
equipment
(9 out of

Entropy 2020,22, 1407 15 of 21
21), and
agriculture
(7 out of 21) sectors. From the export point of view, we observe that the major
suppliers of the US economic sectors are (by number of ingoing links)
chemicals
(13 out of 21 economic
sectors are supplied by
chemicals
sector),
metal prod.
(13 out of 21),
elec/gas/water
(10 out of 21),
and
metals
(9 out of 21). The
chemicals
sector seems to play an important role since it is an economic
hub using products of many other economic sectors and being a supplier of many other economic
sectors. From both the import and export pictures, we observe that the manufacture of equipment
sectors,
com. equipment
(radio, television and communication equipment) and
medical
(medical,
precision and optical instruments, watches and clocks) are linked to other economic sectors only
by hidden links, i.e., in WNEA there is no direct commodities exchange between these sectors and
the others.
agriculture
0Maximum number
of ingoing links
agriculture
agriculture
agriculture
agriculture
agriculture
mining
mining
mining
mining
mining
mining
food
food
food
food
food
food
textile
textile
textile
textile
textile
wood
wood
wood
wood
wood
wood
paper
paper
petroleum
petroleum
petroleum
petroleum
petroleum
chemicals
chemicals
chemicals
chemicals
plastics
plastics
plastics
minerals
minerals
minerals
minerals
minerals
metals
metals
metal prod.
metal prod.
metal prod.
metal prod.
metal prod.
equipment
equipment
office mach.
equipment
com.
equipment
com.
equipment
com.
equipment
com.
equipment
com.
medical
medical
medical
medical
medical
medical
motor
motor
motor
motor
trans. equip.
furniture
furniture
furniture
furniture
furniture
electrical
mach.
electrical
mach.
electrical
mach.
electrical
mach.
electrical
mach.
elec
gas
water
elec
gas
water
textile
trans. equip.
equipment
metals
elec
water
gas
trans. equip.
trans. equip.
trans. equip.
trans. equip.
paper
paper
paper
paper
motor
motor
metals
office mach.
office mach.
office mach.
elec
water
gas
elec
water
gas
elec
water
gas
plastics
plastics
plastics
equipment
equipment
equipment
chemicals
chemicals
metals
metals
minerals
furniture
com.
equipment
petroleum
metal prod.
electrical
mach.
office mach.
office mach.
Figure 9.
Reduced networks of economic sectors of the USA (
top row
), RUS (
central row
), and CHN
(
bottom row
) obtained from the corresponding import reduced Google matrices
GR
(
left panel
) and
export reduced Google matrices
G∗
R
(
right panel
) for year 2009. For each country, the reduced networks
were computed for a set of 21 major economic sectors. From each of them we drew the four strongest
outgoing links. Node labels are sector codes from Table 1. The color of a node corresponds to its
number of ingoing links from 0 (blue color) to the maximum (red color). We distinguish by the blue
color hidden links from direct links present in the raw data. Red bars represent source-side of the
links and arrows represent target-side of the links. The networks have been plotted with radial plot
algorithm in Cytoscape software [42] with manual layout optimization.

Entropy 2020,22, 1407 16 of 21
In Figure 9(middle row), we present the reduced network of the Russian economic sectors for
import (left panel) and export (right panel) exchanges. Here the major importers are the following
economic sectors:
mining
(18 out of 21),
elec/gas/water
(12 out of 21),
food
(11 out of 21). We note that
the
mining
sector uses products of almost all the 21 considered sectors. From the export point of view,
the major exporters are the sectors of
elec/gas/water
(21 of 21),
metals
(13 out of 21),
petroleum
(13 out
of 21),
chemicals
(12 of 21), and
agriculture
(9 out of 21). We note that the
elec/gas/water
sector, which
exploits products of all the other economic sectors, is very central in the Russian economy since it
constitutes the major economic hub. From both import and export pictures, as in the US economy,
the
medical
and
com. equipment
sectors are linked to the other by hidden links. In addtion to these
sectors, the
electrical mach.
(electrical machinery),
metal prod.
(fabricated metal products),
trans.
equip. (transport equipment) sectors also intervene through hidden links.
In Figure 9(bottom row), we present the reduced network of Chinese economic sectors for import
(left panel) and export (right panel) exchanges. The major importers are the sectors of
equipment
(11 out of 21),
chemicals
(10 out of 21),
metals
(10 out of 21),and
mining
(8 out of 21). The major
exporters are the sector of
chemicals
(12 out of 21), and
elec/gas/water
(10 out of 21). As in the US
economy, the chemicals sector is an economic hub playing a central role in the Chinese economy.
3.6. Sensitivity of the EU Countries to the Petroleum Products Price
The combination of the WNEA data and the REGOMAX approach allow us to study the
sensitivity of the country balance to a specific economy sector. In recent studies of the WTN from
the EU COMTRADE database [
22
], such a sensitivity has been determined for the 27 EU countries
(EU members in 2013) in respect to petroleum price variation. Here, for comparison, we show the
balance sensitivity of the same 27 EU countries in respect to the price variation of the petroleum.
The results are presented in Figure 10 for the
petroleum
sectors of USA, Russia, Norway and Saudi
Arabia in 2008. We see that the most sensitive country to US, Russian, and Norwegian petroleum is
Greece while the most sensitive to Saudian petroleum are Greece and Spain. Globally, the influence of
the USA and Russia are comparable while the influences of Norway and Saudi Arabia (SAU) are by a
factor 2–3 smaller.
We note that color maps of the EU balance sensitivities to petroleum products from USA, RUS,
and SAU are somewhat different from the one obtained for the WTN case shown in [
22
] (Figure 6
middle row panels). We attribute this difference to the fact that the
petroleum
sector contains different
petroleum related ISIC products while for the WTN only the petroleum product was considered.
In addition, WNEA comprises real inter-sector and inter-country economic exchanges. We nevertheless
note that the petroleum sensitivity of Netherlands is in any case moderate to strong as in the WTN
study [22].
In Figure 10, we observe that Sweden, Finland, and Latvia are the less sensitive to petroleum
products of any of the considered suppliers. In addition to these countries, we see that the less sensitive
to petroleum products from US are France and Germany, from Russia are Austria, Slovenia and Ireland,
from Norway is Italy, and from Saudi Arabia is Germany. In addition to Greece which is the most
sensitive country to petroleum products for any of the considered suppliers, the most sensitive are
Denmark to US petroleum products, and Spain to Saudian petroleum products.

Entropy 2020,22, 1407 17 of 21
Figure 10.
Balance sensitivity of the 27 EU countries in 2008 to export variation of the
petroleum
sector
of USA (
top left
), RUS (
top right
), Norway (NOR) (
bottom left
) and Saudi Arabia (SAU) (
bottom
right). Color categories are obtained using the Jenks natural breaks classification method [43].

Entropy 2020,22, 1407 18 of 21
4. Discussion
In this work, we apply the reduced Google matrix (REGOMAX) analysis to the World Network
of Economic Activities (WNEA) data in order to determine the interdependence of the economy
activity sectors for several countries with the main accent on USA, Russia, and China. There are
similarities and significant differences for the interactions of the economy sectors of the selected
countries. All the three countries exhibit strong interdependence between
agriculture
and
food
sectors,
that is rather natural since all people need agriculture development for food productions, and also
between
mining
and
petroleum
sectors, that is also very natural. For the US economy, we note that
there are also strong interdependence between
metals
and
metal prod.
sectors, and between
plastics
,
textile
, and
chemicals
sectors. For the Chinese economy, we observe additional interdependence
between
com. equipment
and
office mach.
, and
electrical mach.
to
metals
. From the constructed
reduced networks of economic sectors, for each considered economy, we have determined an economic
hub which uses a broad variety of products from the other economic sectors and supplies also many of
them. For the US economy, the
chemicals
sector is clearly an economic hub. For the Russian economy,
the
elec/gas/water
sector is central as it is the main exporters to all the other sectors and in return
this sector also consumes many resources from the other sectors. For the Chinese economy, as for
the US, the
chemicals
sector is an economic hub. We also determine the sensitivity of the sectors of a
given country to the variation of the price of a specific sector. Globally, for any of the top importer
countries according to PageRank algorithm applied to the WNEA (USA, RUS, CHN, DEU, FRA,
ITA, GBR, JAP, KOR, IND), we observe a strong sensitivity of the
chemicals
,
metals
,
elec/gas/water
,
and
food
sectors to the price increase of the
petroleum
sector products. Contrarily, the
electrical mach.
,
com. equipment
,
trans. equip.
, and
wood
sectors are the most insensitive to the
petroleum
sector.
We compute also the sensitivities of the economic sectors to
chemicals
,
metals
,
motor
, and
mining
sectors and we determine the color map of the EU countries balance sensitivities to a price increase of
products from US, Russian, Norwegian, and Saudian petroleum sectors.
Our study demonstrates that the REGOMAX method allows us to find interdependencies between
economy sectors for selected countries. The WNEA data of OECD-WTO contains transformations
of production of one sector to another that is absent for multiproduct trade data of COMTRADE.
Thus, it would be very desirable to extend OECD-WTO data for more sectors and more recent years.
We hope that this will happen in future years.
Author Contributions:
These authors contributed equally to this work. All authors have read and agreed to the
published version of the manuscript.
Funding:
This research was funded by the Pogramme Investissements d’Avenir ANR-11-IDEX-0002-02,
reference ANR-10-LABX-0037-NEXT, NANOX
N◦
ANR-17-EURE-0009 (THETRACOM and MTDINA projects),
ANR-15-IDEX-0003, ISITE-BFC (GNETWORKS project), and by the Bourgogne Franche-Comté region council
(APEX project).
Acknowledgments:
We thank Hubert Escaith for useful discussions. We had access to the HPC resources of
CALMIP (Toulouse) under the allocation 2018-P0110.
Conflicts of Interest: The authors declare no conflict of interest.

Entropy 2020,22, 1407 19 of 21
Appendix A
-0.012
-0.01
-0.008
-0.006
-0.004
-0.002
0
-0.012
-0.01
-0.008
-0.006
-0.004
-0.002
0
agriculture
mining
food
textile
wood
paper
petroleum
chemicals
plastics
minerals
metals
metal prod.
equipment
office mach.
electrical mach.
com. equipment
medical
motor
trans. equip.
furniture
elec/gas/water
agriculture
mining
food
textile
wood
paper
petroleum
chemicals
plastics
minerals
metals
metal prod.
equipment
office mach.
electrical mach.
com. equipment
medical
motor
trans. equip.
furniture
elec/gas/water
Figure A1.
Same as Figure 7but with same color scale for both panels. Sector balance sensitivity to
petroleum
sector for year 1995 (left) and 2008 (right); horizontal axis represents the sector index in data
order; vertical axis represents the country index in PageRank order for the given year. For each couple
(s
,
c)
we modify the link from (
petroleum
,
c
) towards
(s
,
c)
and compute the
(s
,
c)
balance sensitivity,
D(cpetroleum
,
cs)
. For each plot, a given color corresponds to a given intensity of the sensitivity
D(cpetroleum,cs). Grey column represents self sensitivity (not shown).
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