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Geopolitical interactions from reduced Google

matrix analysis of Wikipedia

Samer El Zant

Institut de Recherche

en Informatique de Toulouse

Université de Toulouse, INPT

Email: samer.elzant@enseeiht.fr

Katia Jaffrès-Runser

Institut de Recherche

en Informatique de Toulouse

Université de Toulouse, INPT

Email: kjr@enseeiht.fr

Dima L. Shepelyansky

Laboratoire de Physique Théorique

du CNRS, IRSAMC

Université de Toulouse, INPT

Email: dima@irsamc.ups-tlse.fr

Abstract—Interactions between countries originate from di-

verse aspects such as geographic proximity, trade, socio-cultural

habits, language, religions, etc. Geopolitics studies the inﬂuence

of a country’s geographic space on its political power and its

relationships with other countries. This work reveals the potential

of Wikipedia mining for geopolitical study. Actually, Wikipedia

offers solid knowledge and strong correlations among countries

by linking web pages together for different types of information

(e.g. economical, historical, political, and many others). The ma-

jor ﬁnding of this paper is to show that meaningful results on the

inﬂuence of country ties can be extracted from the hyperlinked

structure of Wikipedia. We leverage a novel stochastic matrix

representation of Markov chains of complex directed networks

called the reduced Google matrix theory. For a selected small size

set of nodes, the reduced Google matrix concentrates direct and

indirect links of the million-node sized Wikipedia network into

a small Perron-Frobenius matrix that preserves the PageRank

probabilities of the global Wikipedia network. We perform a

novel sensitivity analysis that leverages this reduced Google

matrix to characterize the inﬂuence of relationships between

countries from the global network. We apply this analysis to

the set of 27 European Union countries. We show that with

our sensitivity analysis we can exhibit easily very meaningful

information on geopolitics from ﬁve different Wikipedia editions

(English, Arabic, Russian, French and German).

I. INTRODUCTION

Relationships between countries have always been of utmost

interest to study for countries themselves as they have to be

accounted for into any country’s strategic and diplomatic plan.

Studies are driven by observing the inﬂuence of a relationship

between two countries on other countries from different per-

spectives listing economic exchanges, social changes, history,

politics, religious, martial, regional as seen in [1]. The major

ﬁnding of this paper is to show that meaningful results on

geopolitics interactions could be extracted from Wikipedia for

a given selection of countries. Therefore, it can be leveraged

to provide a picture of countries relationships offering a

new framework for long-term geopolitical studies. In [2], S.

Javanmardi et al. show that even though anyone can edit a

Wikipedia entry at any time, the average article quality in-

creases as it goes through various edits. Wikipedia’s accuracy

for its scientiﬁc entries has been proved by comparing it to

Encyclopedia Britannica and to PDQ - NCI’s Comprehensive

Database in [3], [4]. To sum up, Wikipedia has become the

largest accurate reliable free online open source of knowledge.

TABLE I

List of EU countries.

Wikipedia edition English French German

Countries CC Color K K K

France* FR BL 1 1 2

United Kingdom* GB GN 2 4 24

Germany DE BL 3 2 1

Italy IT BL 4 3 4

Spain* ES OR 5 5 5

Poland* PL RD 6 8 6

Netherlands NL BL 7 7 7

Sweden* SE PK 8 11 8

Romania RO RD 9 18 17

Belgium BE BL 10 6 9

Austria AT PK 11 9 3

Greece GR OR 12 13 14

Portugal PT OR 13 12 11

Ireland IE GN 14 19 16

Denmark DK GN 15 14 10

Finland FI PK 16 17 15

Hungary HU RD 17 10 13

Czech Republic CZ RD 18 15 12

Bulgaria BG RD 19 20 20

Estonia EE RD 20 24 22

Slovenia SI RD 21 23 23

Slovakia SK RD 22 16 18

Lithuania LT RD 23 22 21

Cyprus CY RD 24 27 27

Latvia LV RD 25 25 25

Luxembourg LU BL 26 21 19

Malta MT RD 27 26 26

PageRank Kfor EnWiki, FrWiki and DeWiki. Color code groups EU

countries into 5 subsets: Blue (BL) for Founders, Green (GN) for 1973 new

member states, Orange (OR) for 1981 to 1986 new member states, Pink

(PK) for 1995 new member states and Red (RD) for 2004 to 2007 new

member states. Standard country codes (CC) are given as well. Countries in

bold are the selected ones for each group.

Unique to Wikipedia is that articles make citations to each

other, providing a direct relationship between webpages. As

such, Wikipedia generates a larger directed network of article

titles with a rather clear meaning. For these reasons, it is

interesting to apply algorithms developed for search engines

of World Wide Web, those like the PageRank algorithm

[5], to analyze the ranking properties and relations between

Wikipedia articles. For various language editions of Wikipedia

it was shown that the PageRank vector produces a reliable

ranking of historical ﬁgures over 35 centuries of human history

[6]–[9] and a solid Wikipedia ranking of world universities

(WRWU) [6], [11]. It has been shown that the Wikipedia

ranking of historical ﬁgures is in a good agreement with

the well-known Hart ranking [12], while the WRWU is in a

good agreement with the Shanghai Academic ranking of world

universities [13].

This paper analyses the networks of articles extracted from

5 language editions of Wikipedia to study the inﬂuence of

countries on each other. Previous work [16] has identiﬁed

the strongest ties between countries, but this one focuses on

capturing the impact of a change in the strength of a relation-

ship between two countries on the overall network interactions

of selected countries via the global network. The impact on

the overall network structure is measured by calculating the

variation of importance of the nodes in the network. We

show that this sensitivity analysis renders a reasonable and

meaningful idea of the inﬂuence of a given bilateral tie on the

whole network.

We have conducted our geopolitics study for the target set of

27 European Union member states. As such, from the global

network of articles of Wikipedia we have derived the reduced

Google matrix GRfor these 27 EU states. Thus, GRreﬂects

in a 27-by-27 matrix the complete (direct and indirect) rela-

tionships between countries. To quantify the relative inﬂuence

of one relationship between two nations on all other nations,

we propose in this paper to compute a logarithmic derivative

of the PageRank probabilities calculated from GRand ˜

GR.

PageRank probabilities are derived from GRas explained later.

They represent the importance of a node in the global network

of articles. ˜

GRis almost equal to GR. It only differs by the

values of one column. If the relationship going from nation jto

nation i, only the values of column jare changed to relatively

inﬂate the probability ˜

GR(i, j)of nation jending in nation

icompared to the other ones. This is done in practice by

modifying ˜

GR(i, j)and then normalizing the column again to

unity to enforce the column normalization property of Google

matrices. Results are derived for 5 different Wikipedia editions

(Data collected February 2013) from the set of 24 analyzed in

[9]: EnWiki, ArWiki, RuWiki, DeWiki and FrWiki that contain

4.212, 0.203 , 0.966, 1.533 and 1.353 millions of articles each.

The selected countries are the 27 EU countries as of February

2013 (Croatia joined in July 2013) as mentioned in Table I.

The paper is organized as follows. At ﬁrst we introduce

the reduced Google matrix theory, together with a primer

on Google Matrix and PageRank calculations. The reduced

Google matrix is illustrated for the 27 EU set of states.

Next, the methodology for our link sensitivity analysis is

presented. A detailed analysis of EU countries is given in

the Results section, with a special focus on the sensitivity

analysis of important relationships among member states.

Finally, conclusions are drawn in the last section.

II. RE DU CE D GOOGLE MATRI X TH EO RY

It is convenient to describe the network of NWikipedia ar-

ticles by the Google matrix Gconstructed from the adjacency

matrix Aij with elements 1if article (node) jpoints to article

(node) iand zero otherwise. Elements of the Google matrix

take the standard form Gij =αSij + (1 −α)/N [5], [10],

where Sis the matrix of Markov transitions with elements

Sij =Aij /kout(j),kout (j) = PN

i=1 Aij 6= 0 being the node

jout-degree (number of outgoing links) and with Sij = 1/N

if jhas no outgoing links. The damping factor 0< α < 1is

which for a random surfer determines the probability (1−α)to

jump to any node; below we use the standard value α= 0.85.

The right eigenvector of Gwith the unit eigenvalue gives the

PageRank probabilities P(j)to ﬁnd a random surfer on a node

j. We order nodes by decreasing Pgetting them ordered by

the PageRank index K= 1,2, ...N with a maximal probability

at K= 1. From this global ranking we capture the top local

PageRank mentioned in Tab. I.

Reduced Google matrix is constructed for a selected subset

of nodes (articles) following the method described in [14]–[16]

and based on concepts of scattering theory used in different

ﬁelds of mesoscopic and nuclear physics or quantum chaos.

It captures in a Nr-by-NrPerron-Frobenius matrix the full

contribution of direct and indirect interactions happening in

the full Google matrix between the Nrnodes of interest.

In addition the PageRank probabilities of selected Nrnodes

are the same as for the global network with Nnodes, up

to a constant multiplicative factor taking into account that

the sum of PageRank probabilities over Nrnodes is unity.

Elements of reduced matrix GR(i, j)can be interpreted as the

probability for a random surfer starting at web-page jto arrive

in web-page iusing direct and indirect interactions. Indirect

interactions refer to paths composed in part of web-pages

different from the Nrones of interest. Even more interesting

and unique to reduced Google matrix theory, we show here that

intermediate computation steps of GRoffer a decomposition

of GRinto matrices that clearly distinguish direct from indirect

interactions: GR=Grr +Gpr +Gqr [15]. Here, Grr is the sub-

matrix of Grepresenting the original direct links between the

selected Nrnodes. Fig. 1 shows that Gpr is rather close to the

matrix in which each column is given by the PageRank vector

Pr, ensuring that PageRank probabilities of GRare the same

as for G(up to a constant multiplier). As such, Gpr doesn’t

provide relevant information to characterize the importance

of links between the selected nodes. The one playing an

interesting role is Gqr, which captures the effect of all indirect

paths connecting the selected Nrnodes in the full network of

Nnodes (see [14]–[16]). The matrix Gqr =Gqrd +Gqrnd

has diagonal (Gqrd) and non-diagonal (Gqrnd ) parts. Gqrnd is

leveraged for the studies of Section III-B. Results of sections

IV and V are based on GRonly. The complete theoretical

background is to be found in [14]–[16].

III. RES ULTS :GRPROPERTIES

A. Reduced Google matrix of 27 EU set.

As an example, we have picked the EnWiki edition to plot

the matrices GR,Gpr,Grr , and Gqrnd in Fig. 1. As GRis

per-column normalized and dominated by the projector Gpr

contribution, which is proportional to the global PageRank

Fig. 1. Density plots of GRand its decomposition for 27 EU extracted

from EnWiki. GR(top left), Gpr (top right), Grr (bottom left) and Gqrnd

(bottom right). Max values in red (0.14 in top panels; 0.003 in bottom left;

0.008 in bottom right), intermediate in green and min (≈0) in blue.

probabilities (more details in [14], [15]), this prevents a

meaningful per-line analysis. Grr provides information only

on direct links between countries as it lists the genuine Google

matrix probability for a random surfer to jump from node jto

i. On the contrary, Gqrnd offers a much more uniﬁed view of

countries interactions as it highlights more general indirect (or

hidden) interactions views via the rest of nodes. It captures the

contribution of all indirect paths connecting two nodes iand j

in the full network of Wikipedia articles. For the three selected

languages editions, we have identiﬁed very strong hidden links

connecting Finland to Sweden. Other interesting hidden links

are between Ireland and United Kingdom in DeWiki or in

EnWiki, the hidden links connecting Luxembourg to France.

B. Networks of friends

As proposed in [16], it is possible to extract from Gqrnd a

network of friendships to easily illustrate hidden links in the

network. For the sake of simplicity, we refer next to Gqrnd

using Gqr notation. To create these networks of friends, we

divide the set of Nrnodes into representative groups as shown

in Tab I. EU countries are grouped upon their accession date

to the union. One leading country per EU member state group

has been selected as well.

For each leading country j, we extract from Gqr the top 4

Friends given by the 4 best values of the elements of column j.

In other words, it corresponds to destinations of the 4 strongest

outgoing links of j. These networks of top 4 friends have been

calculated for the ﬁve editions of Wikipedia. Top 4 friends of

EU leading countries are plot on the graphs of Fig. 2. Results

for EnWiki, FrWiki and DeWiki are presented here. The black

thick arrows identify the top 4 friends interactions. Red arrows

represent the friends of friends interactions that are computed

recursively until no new edge is added to the graph. All graphs

are visualized with Yifan Hu layout [17] using [18].

It can be noticed in Fig. 2 that the order of arrival of member

states is meaningful. Indeed, nodes of the same color are

closely interconnected. It is worth noting as well that Germany,

Fig. 2. Relationship structure extracted from Gqrnd for the network of

EU countries. friends induced by the leading countries (FR, GB, ES, SE,

PL) of each group. Results are plotted for EnWiki (left), FrWiki (middle) and

DeWiki (right). Node colors represent geographic appartenance to a group of

countries (cf. Table I for details). Selected countries points with a bold black

arrow to its top 4 friends. Red arrows show friends of friends interactions

computed until no new edges are added to the graph.

TABLE II

Cross-edition friends extracted from Gqr of EU countries.

Top Gqr Wiki friends present in

country all 5 editions 4 out of 5 editions 3 out of 5 editions

FR BE -ES IT

GB IE DK - FR

ES IT - PT FR BE

SE DK - FI EE

PL CZ DE - HU - LT - SK

as one of the Founders, bridges the group of Founders to

Sweden (the leader of the countries that have joined EU in

1995) and Poland (the leader of the countries that have joined

EU between 2004 and 2007) in FrWiki and EnWiki. From

EnWiki and DeWiki, strong ties are seen between Italy and

France, while it is not the case for FrWiki authors. This is

an example of cultural bias. However, lots of links are to

seen in all three editions: GB-IE, SE-FI, ES-PT, PL-LT, IT-

GR and many others. In all editions, Benelux and Nordic

countries create a cluster densely interconnected. To underline

this constant presence of links, we give in Table II the list of

friends that are among the top 4 ones in all 5 editions, in 4

out of 5 and in 3 out of 5. For each leading country, around

2 to 3 top friends are present across all editions.

IV. LINK SENSITIVITY ANALYSIS

A. Inﬂuence analysis of geopolitical ties using GR

Previous developments in this article show that GRcaptures

essential interactions between countries. These interactions are

extracted from Wikipedia and thus stem from all links covering

this very rich network of webpages.

The point is now to see how some ties between countries

inﬂuence the whole network structure. More speciﬁcally, we

focus here on capturing the impact of a change in the strength

of a relationship between two countries on the importance

of the nodes in the network. Therefore we have designed a

sensitivity analysis that measures a logarithmic derivative of

the PageRank probability when the transition probability of

only one selected link is increased for a speciﬁc couple of

nodes in GR, relatively to the other nodes.

Our sensitivity analysis is performed for a directed link

where the relationship going from country ito jis increased.

We investigate in the last part of this Section the imbalance

between the inﬂuence of two opposite direction interactions.

In other words, we conduct the aforementioned sensitivity

analysis for the link going from country ito j, and for the

link going in the opposite direction from jto i. For each

pair of countries, we derive from this two-way sensitivity the

relationship imbalance to identify the most important player

in the relationship.

B. Sensitivity analysis

We deﬁne δas the relative fraction to be added to the

relationship from nation jto nation iin GR. Knowing δ, a new

modiﬁed matrix ˜

GRis calculated in two steps. First, element

˜

GR(i, j)is set to (1 + δ)·GR(i, j ). Second, all elements of

column jof ˜

GRare normalized to 1 (including element i)

to preserve the unity column-normalization property of the

Google matrix. Now ˜

GRreﬂects an increased probability for

going from nation jto nation i.

It is now possible to calculate the modiﬁed PageRank

eigenvector ˜

Pfrom ˜

GRusing the standard ˜

GR˜

P=˜

Prelation

and compare it to the original PageRank probabilities Pcal-

culated with GRusing GRP=P. Due to the relative change

of the transition probability between nodes iand j, steady

state PageRank and CheiRank probabilities are modiﬁed. This

reﬂects a structural modiﬁcation of the network and entails a

change of importance of nodes in the network. These changes

are measured by a logarithmic derivative of the PageRank

probability of node agiven by:

D(j→i)(a) = (dPa/dδij )/Pa= ( ˜

Pa−Pa)/(δij Pa)(1)

Notation (j→i)indicates that the link from node jto node i

has been modiﬁed. Element D(j→i)(a)gives the logarithmic

variation of PageRank probability for country aif the link from

jto ihas been modiﬁed. We will refer to this variation as the

sensitivity of nation ato the relationship from nation ito nation

j. If this sensitivity is negative, country ihas lost importance in

the network. On the opposite, a positive sensitivity expresses

a gain in importance. The computation has been tested for

values of δ=±0.01,±0.03,±0.05. The result is not sensitive

to δand following results are given for δ= 0.03.

C. Relationship imbalance analysis

As introduced earlier, sensitivity D(j→i)(a)of Eq (1) mea-

sures the change of importance of country aif the link from

nation jto ihas been changed. The sensitivity of node ato

a change in one direction is not necessarily the same as its

sensitivity to the change in the opposite direction. We deﬁne

as such the 2-way sensitivity of node awhich is simply the

sum of the sensitivities calculated for both directions:

D(i↔j)(a) = D(i→j)(a) + D(j→i)(a)(2)

The two-way sensitivity can be leveraged to ﬁnd out, for a

pair of countries aand b, which one has the most inﬂuence

on the other one. Therefore, we deﬁne the following metric :

F(a, b) = D(a↔b)(a)−D(a↔b)(b)(3)

Here, we measure the 2-way sensitivity for nodes aand b

when the link between them is modiﬁed both ways in GR.

If F(a, b)is positive, it means that the 2-way sensitivity of

ais larger than the 2-way sensitivity of b. In this case, ais

more inﬂuenced by bthan bby a. We can say that bis the

strongest country. If F(a, b)is negative, we can say that ais

the strongest country.

V. SENSITIVITY ANALYSIS RESULTS

The sensitivity analysis previously presented has been per-

formed for the 27 EU reduced network with 3 Wikipedia

editions: EnWiki, FrWiki and DeWiki. This analysis calculates

for each directed link j→iof the reduced 27 EU network

the sensitivity Dj→i(a)of each country a. From this, the

relationship imbalance analysis has been calculated as well for

each pair of nations. Note that if the modiﬁed link is clearly

identiﬁed in the following, we will drop the index i→jin

our sensitivity measure notation for clarity.

In order to better capture the countries sensitivities from

a multicultural perspective, all sensitivity results are averaged

over the three editions using ¯

D=1

3P3

i=1 Di, with ithe index

of a Wikipedia edition.

A. Sensitivity analysis

We start this analysis by introducing a ﬁrst simple example

where Italy increases its relationship with France. Then, we

analyze the impact on the EU countries of Great Britain’s

exit (i.e. Brexit) from European Union. Next, we highlight

the sensitivity of Luxembourg to the increase of Germany

and France’s cooperation with other member states. Finally,

we present the results that underline the strong ties that exist

between groups of countries that function together in Europe.

For each sensitivity analysis, we show an axial represen-

tation of the sensitivity ¯

D(cf. Fig. 5, Fig. 6, Fig. 3, Fig. 4).

Each axis represents the sensitivity values obtained for a given

link variation.

1) Great Britain ties to France and Germany: The United

Kingdom has triggered article 50 on March 27, 2017 to leave

the European Union as a consequence of the referendum

of June 23rd, 2016 [23]. To understand its impact on EU

countries with our dataset, we have reduced (and not increased

as done in other studies) the GRtransition probability UK

towards France or Germany. We remind that our network

is dated by 2013 but it captures the strong UK inﬂuence.

Results are shown in Fig. 3 and indicate that Ireland and

Cyprus are by far the most negatively affected countries in

both cases. Moreover, the sensitivity of UK is negative as it

Fig. 3. Axial representation of ¯

Dfor link modiﬁcations from {GB} to

{FR or DE}. (A): GB to FR. (B): GB to DE.

Fig. 4. Axial representation of ¯

Dfor link modiﬁcations from {FR or

DE} to {GB or IT}. (A): FR to GB. (B): DE to GB. (C) FR to IT.

beneﬁts less from France’s or Germany’s inﬂuence. These facts

have been recently backed up by specialists. In [24], a study

delivered by the London School of Economics discussing the

consequences of Brexit forecasts that UK will loose 2.8% of

its Gross domestic product (GDP)1. Similarly, [24] shows that

Ireland will loose as well 2.3% of its GDP, which is the largest

proportional loss caused by Brexit. Cyprus-UK Relations are

strong as claimed by the ofﬁcial website of the Ministry of

Foreign Affairs of Cyprus [26]. Referring to [20], UK is the

4th top export destination for Cyprus with $242M and the 2nd

import origin with $508M. As such, this clear bond of UK with

Cyprus explains that if GB suffers from Brexit, Cyprus will

do as well. Our data strikingly exhibits the same conclusion

as shown in Fig. 3.

2) Luxembourg’s sensitivity to Germany and France:

Luxembourg shares its borders with Belgium, Germany and

France with whom it has strong and diverse relationships.

Luxembourg has an open economy. Together with Belgium,

they position themselves as the 12th largest economy in the

world. Two of the top three export and import countries

of Belgium-Luxembourg are Germany ($44.6B, $50.4B) and

France ($43.8B, $36.8B) [20]. Ofﬁcial languages in Luxem-

bourg are Luxembourgish, French and German. Luxembourg

has robust relationships with France [27], [29] and Ger-

many [30] in various areas such as ﬁnance, culture, science,

security or nuclear power. It is clear that Luxembourg will

suffer if one of these EU countries reduces its exchanges

with it. In Fig. 4, we clearly show that Luxembourg is

strongly inﬂuenced by France and Germany. If France or

Germany increases its relationships with Italy or Great Britain,

Luxembourg is by far the most impacted country.

3) Clusters of countries: By analyzing the sensitivity of

countries to various 2-nation relationships, we have noticed

that several groups of nations function together. These groups

are strongly interconnected, and if anyone of these group

members increases its relationship strength with a country

outside of the group, all group members loose importance in

the network. We highlight two meaningful examples next: the

cluster of Nordic countries and the cluster Austro-Hungarian

cluster. Other clusters we have identiﬁed in our network are

for instance the cluster of Benelux countries (e.g. Belgium,

the Netherlands and Luxembourg) or the cluster of the Iberian

1GDP: monetary value of all the ﬁnished goods and services produced

within a country’s borders in a speciﬁc time period [25].

Fig. 5. Axial representation of ¯

Dfor link modiﬁcations from Nordic

countries to {FR or DE}. (A): DK to DE. (B): SE to DE. (C): FI to DE.

(D): DK to FR. (E): SE to FR.

peninsula (e.g. Portugal and Spain).

For both investigated groups, we test the inﬂuence of an

increase in collaboration from one member of the group to

France or to Germany. France and Germany have been chosen

as they are central members of European Union.

The Nordic countries Denmark, Finland, and Sweden have

much in common: their way of life, history, language and

social structure [19]. After World War II, the ﬁrst concrete

step into unity was the introduction of a Nordic Passport Union

in 1952. Nordic countries co-operate in the Nordic Council, a

geopolitical forum. In the Nordic Statistical Yearbook [19],

Klaus Munch illustrates that “The Nordic economies are

among the countries in the Western World with the best

macroeconomic performance in the recent ten years”. Nordic

countries should keep cooperating to stay strong. Thus, if

any Nordic country attempts to abandon these relationships

in favor of other countries, it will negatively impact the

remaining Nordic countries. Our sensitivity analysis illustrates

this impact in Fig. 5. In these ﬁgs, we show how the relation-

ship increase between any Nordic country towards France or

Germany induces a drop in sensitivity for Nordic countries.

Referring to [21], relations between Slovenia, Hungary

and Austria are tight. Hungary has supported Slovenia for

its NATO membership applications and Austria has assisted

Slovenia in entering European Union. Relationships between

Austria and Hungary are important for both countries in

the economic, political and cultural ﬁelds [22]. Concerning

economy [20], Austria is one of the top import origins for

Hungary and Slovenia with $5.54B and $2.37B respectively.

Similarly to the Nordic group of countries, if Austria, Slovenia

or Hungary increases its relationships with another European

country, the other two will be affected. Sensitivity analysis

backs up this statement as seen in Fig. 6.

B. Relationship imbalance analysis

Relationship imbalance analysis has been derived for all

pairs of European countries following Eq (3). Fig. 7 shows a

density plot of F(a, b). We recall that if F(a, b)is negative,

nation ahas more inﬂuence on nation bthan bon a. If F(a, b)

Fig. 6. Axial representation of ¯

Dfor link modiﬁcations from {AT, HU

and SI} to {FR or DE}. (A): AT to FR. (B): HU to FR. (C): SI to FR. (D):

AT to DE. (E): HU to DE. (F): SI to DE.

is positive, nation bdominates nation a. According to The

Globe of Economic Complexity [28] and identical to our results

in Fig. 7, Germany and France are the two largest economies in

Europe. From GRwe can clearly see the dominance of France

and Germany on other EU countries. Another interesting result

of Fig. 7 is the equal inﬂuence between all pairs of countries

created by one member of {GR, PT, IE, DK, FI, HU} and

another of {BG, EE, SI, SK, LT, CY, LV, LU, MT}. These

pairs have F(a, b)close to zero and are plotted with orange

color in Fig. 7.

Fig. 7. Relationship imbalance analysis: F-representation for 27 EU

network. X-axis and Y-axis represent aand brespectively.

VI. CONCLUSION

This work offers a new perspective for future geopolitics

studies. It is possible to extract from multi-cultural Wikipedia

networks a global understanding of the interactions between

countries at a regional scale. Reduced Google matrix the-

ory has been shown to exhibit hidden interactions among

countries, resulting in new knowledge on geopolitics. Results

show that our sensitivity analysis captures the importance of

relationships on network structure. This analysis relies on

the reduced Google matrix and leverages its capability of

concentrating all Wikipedia knowledge in a small stochastic

matrix. We stress that the obtained sensitivity of geopolitical

relations between two countries and its inﬂuence on other

world countries is obtained on a pure mathematical statistical

analysis without any direct appeal to political, economical and

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