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< \newcommand\GR{{G_{\rm R}}}
< \newcommand\GRprime{{\tilde{G}_{\rm R}}}
< \newcommand\Pprime{{\tilde{P}}}
< \newcommand\Grr{{G_{rr}}}
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< 
< \begin{document}
< 
< \title*{Analysis of world terror networks from \\ the reduced Google matrix of Wikipedia}
< % Use \titlerunning{Short Title} for an abbreviated version of
< % your contribution title if the original one is too long
< \author{Samer El Zant,  Klaus M. Frahm, Katia Jaffr\`es-Runser, Dima L. Shepelyansky}
< % Use \authorrunning{Short Title} for an abbreviated version of
< % your contribution title if the original one is too long
< \institute{Samer El Zant \at Institut de Recherche en Informatique de Toulouse/INPT, Universit\'e de Toulouse, Toulouse, France, \email{samer.elzant@enseeiht.fr}
< \and Klaus M. Frahm \at Lab. de Phys. Theorique, CNRS, Univ. de Toulouse, CNRS, UPS, 
< Toulouse, France, \email{frahm@irsamc.ups-tlse.fr}
< \and Katia Jaffr\`es-Runser \at Institut de Recherche en Informatique de Toulouse/INPT, Universit\'e de Toulouse, Toulouse, France, \email{kjr@enseeiht.fr}
< \and Dima L. Shepelyansky \at Lab. de Phys. Theorique, CNRS, Univ. de Toulouse, CNRS, UPS, 
< Toulouse, France, \email{dima@irsamc.ups-tlse.fr}
< }
< %
< % Use the package "url.sty" to avoid
< % problems with special characters
< % used in your e-mail or web address
< %
< \maketitle
< 
< 
< \abstract{
< We apply the reduced Google matrix method for analysis 
< of the interactions between 95 terrorist groups
< and determining their relations and influence on 64 world 
< countries. This is done on the basis of the Google matrix of the English Wikipedia
< (2017) containg 5416537 articles which accumulate a great part of global
< human knowledge. The reduced Google matrix takes into account the
< direct and hidden links between selected 159 nodes (articles)
< appearing due to all paths of a random surfer moving over the whole network.
< As a result we obtain the network structure of terrorist groups
< and their relations with selected countries. Using the sensitivity of PageRank
< to a weight variation of specific links we determine the geopolitical
< sensitivity and influence of specific terrorist groups
< on world countries. We argue that this approach can find useful application for
< more extensive and detailed data bases.  
< }
< 
< \section{Introduction}
< \label{sec:1}
< 
< {\it ''A new type of terrorism threatens the world,
< driven by networks of fanatics determined to inflict 
< maximam civilian and economic damages on distant targets in pursuit of 
< their extremist goals''} \cite{sageman1}. The origins of this world wide phenomenon
< are under investigations in political, social and religious sciences
< (see e.g.  \cite{kepel1,kepel2,sageman1,sageman2} and Refs. therein).
< At the same time the number of terrorist groups  is growing in the world \cite{taliban1}
< reaching more than 100 officially recognized groups acting in various
< countries of the world \cite{wikispgroups}. These numbers become rather large and 
< the mathematical analysis of multiple interactions between these groups
< and their relations with world countries becomes of great actuality.
< The first steps in this direction are reported in a few publications 
< (see e.g. \cite{hicks,latora}) showing that the network science methods 
< (see e.g. \cite{dorogovtsev})
< should be well adapted to such type of investigations. However, it is 
< difficult to obtain a clear network structure with all dependencies which
< are emerging from the surrounding world with all its complexity.
< 
< In this work we use the approach of the Google matrix $G$ and PageRank algorithm
< developed by Brin and Page for large scale WWW network analysis \cite{brin}.
< The mathematical and statistical properties of this approach for various networks 
< are described in \cite{langville,rmp2015}. The efficiency of these methods
< are demostrated for Wikipedia and world trade networks in 
< \cite{eomplos,ermannwtn,lages}. For the analysis of the terror networks
< we use the reduced Google matrix approach developed recently 
< \cite{frahm,politwiki,geop}. This approach selects from a global large scale network
< a subset of nodes of interest and constructs the reduced Google matrix 
< $\GR$ for this subset including all indirect links connecting the subset nodes 
< via the global network. The analysis of subsets of political leaders 
< and world countries subsets of Wikipedia networks in various language editions
< demonstrated the efficiency of this analysis \cite{politwiki,geop}. Here, for the 
< English Wikipedia network (collected in May 2017), we choose a subset
< of $N_g = 95$ terrorist groups represented by
< Wikipedia articles about terrorist groups
< listed at least by two countries in \cite{wikispgroups} (see Table~\ref{tab:groups}).
< In addition we select the group of $N_c=64$ related world countries
< given in Table~\ref{tab:countries}. This gives us the size of $\GR$ being $N_r=N_g+N_c=159$
< that is much smaller then the global Wikipedia network with
< $N=5416537$ nodes (articles) and $N_\ell = 122232932$ links generated by 
< quotation links from one article to another. The method of the reduced Google matrix
< and the obtained results for interactions 
< between terrorist groups and countries  are described in the next Sections.
< 
< \section{Reduced Google matrix}
< \label{sec:2}
< 
< It is convenient to describe the network of $N$ Wikipedia articles by the Google matrix $G$ constructed from 
< the adjacency matrix $A_{ij}$ with elements $1$ if article (node) $j$ 
< points to  article (node) $i$ and zero otherwise. 
< In this case, elements of the Google matrix take the standard form 
< $G_{ij} = \alpha S_{ij} + (1-\alpha) / N$ \cite{brin,langville,rmp2015},
< where $S$ is the matrix of Markov transitions with elements  $S_{ij}=A_{ij}/k_{out}(j)$, 
< $k_{out}(j)=\sum_{i=1}^{N}A_{ij}\neq0$ being the node $j$ out-degree
< (number of outgoing links) and with $S_{ij}=1/N$ if $j$ has no outgoing links (dangling node). 
< Here $0< \alpha <1$ is the damping factor  which for a random surfer
< determines the probability $(1-\alpha)$ to jump to any node; below we use the standard value $\alpha=0.85$. 
< The right eigenvector
< of $G$ with the unit eigenvalue gives the PageRank probabilities
< $P(j)$ to find a random surfer on a node $j$. We order all nodes by decreasing probability $P$ 
< getting them ordered by the PageRank index $K=1,2,...N$ with a maximal probability at $K=1$.
< From this global ranking we obtain the local ranking of groups and countries given in 
< Tables~\ref{tab:groups},~\ref{tab:countries}.
< 
< 
< The reduced Google matrix $\GR$ is constructed for a selected subset of 
< nodes (articles) following the method described
< in \cite{frahm,politwiki} 
< %%% some proposition from Klaus:
< and based on concepts of scattering theory 
< used in different fields of mesoscopic and nuclear physics or 
< quantum chaos. 
< %%% maybe some additional citations behind ``chaos'' in the 
< %%% last phrase.
< This matrix has $N_r$ nodes and belongs 
< to the class of Google matrices. In addition the 
< PageRank probabilities of selected $N_r$ nodes are the same 
< as for the global network with $N$ nodes,
< up to a constant multiplicative factor taking into account that 
< the sum of PageRank probabilities over $N_r$
< nodes is unity. The matrix $\GR$ is represented as a sum of three matrices 
< (components)
< $\GR = \Grr + \Gpr + \Gqr$ \cite{politwiki}. 
< The first term $\Grr$ is given by the direct links between selected 
< $N_r$ nodes in the global $G$ matrix with $N$ nodes, 
< the second term $\Gpr$ is rather close to 
< the matrix in which each column is given by 
< the PageRank vector $P_r$ so that it ensures the main
< request that the PageRank probabilities of $\GR$ are 
< the same as for $G$ (up to a constant multiplier).
< Therefore  $\Gpr$ does not provide much information about direct 
< and indirect links between selected nodes.
< The most interesting is the third matrix $\Gqr$ which takes 
< into account all indirect links between
< selected nodes appearing due to multiple links via 
< the global network nodes $N$ \cite{frahm,politwiki}.
< The matrix  $\Gqr = \Gqrd + \Gqrnd$ has diagonal ($\Gqrd$)
< and nondiagonal ($\Gqrnd$) parts. The part $\Gqrnd$
< represents the main interest since it describes indirect interactions between nodes. 
< The explicit formulas as well as the mathematical and numerical computation 
< methods of all three components of $\GR$ are given 
< in \cite{frahm,politwiki,geop}. 
< 
< The selected groups and countries are given in Tables~\ref{tab:groups},~\ref{tab:countries}
< in order of their PageRank probabilities. All countries have PageRank probabilities 
< being larger then those of groups so that they are well separated.
< 
< \section{Results}
< \label{sec:3}
< 
< In this work we extract from $\GR$ a network of 64 countries and 95 groups. 
< This network reflects direct and indirect interactions between countries and groups, 
< which motivates us to study the relative influence of group alliances 
< on the other ones and on the countries. 
< The matrix $\GR$ and its three components $\Grr$, $\Gpr$ and $\Gqr$ 
< are computed for $N_r=165$ Wikipedia network nodes 
< formed by $N_c=64$ country nodes and $N_g=95$ group nodes.  
< The weights of these three $\GR$ components are 
< $\Wrr$=0.0644, $\Wpr$=0.8769 and $\Wqr$=0.0587 (the weight is given by sum of all matrix elements divided by $N_r$,
< thus $\Wrr + \Wqr + \Wqr = 1$). 
< The dominant component is $\Gpr$ but as stated above it is approximately given by
< columns of PageRank vector so that the most interesting information is provided by
< $\Grr$ and especially the component $\Gqr$ given by indirect links \cite{politwiki,geop}.
< 
< % Please add the following required packages to your document preamble:
< % \usepackage[normalem]{ulem}
< % \useunder{\uline}{\ul}{}
< \begin{table}[]
< \centering
< \caption{List of selected terrorist groups (from \cite{wikispgroups}) attributed to 6 categories marked by color.}
< \label{tab:groups}
< %\begin{tabular}{|c|c|c|c|c|c|}
< \begin{tabular}{|p{4cm}|c|c|p{4cm}|c|c|}
< \hline
< Name                                          & KG  & Color                     & Name                                                    & KG  & Color                     \\ \hline
< Islamic State of Iraq and the Levant          & 1  & {\color[HTML]{3531FF} BL} & Hezb-e Islami Gulbuddin                                 & 49 & {\color[HTML]{FE0000} RD} \\ \hline
< Al-Qaeda                                      & 2  & {\color[HTML]{3531FF} BL} & Kach and Kahane Chai                                    & 50 & BK                        \\ \hline
< Taliban                                       & 3  & {\color[HTML]{FE0000} RD} & Palestine Liberation Front                              & 51 & {\color[HTML]{F56B00} OR} \\ \hline
< Provisional Irish Republican Army             & 4  & BK                        & Harkat-ul-Mujahideen                                    & 52 & {\color[HTML]{FE0000} RD} \\ \hline
< Hamas                                         & 5  & {\color[HTML]{F56B00} OR} & Kurdistan Free Life Party                               & 53 & BK                        \\ \hline
< Hezbollah                                     & 6  & {\color[HTML]{F56B00} OR} & Indian Mujahideen                                       & 54 & {\color[HTML]{FE0000} RD} \\ \hline
< Muslim Brotherhood                            & 7  & {\color[HTML]{3531FF} BL} & Abu Nidal Organization                                  & 55 & {\color[HTML]{F56B00} OR} \\ \hline
< Liberation Tigers of Tamil Eelam              & 8  & {\color[HTML]{FE0000} RD} & Hizbul Mujahideen                                       & 56 & {\color[HTML]{FE0000} RD} \\ \hline
< Kurdistan Workers' Party                      & 9  & BK                        & Libyan Islamic Fighting Group                           & 57 & {\color[HTML]{009901} GN} \\ \hline
< Al-Shabaab (militant group)                   & 10 & {\color[HTML]{009901} GN} & Islamic State of Iraq and the Levant in Libya           & 58 & {\color[HTML]{009901} GN} \\ \hline
< ETA (separatist group)                        & 11 & BK                        & Revolutionary People's Liberation Party/Front           & 59 & BK                        \\ \hline
< FARC                                          & 12 & BK                        & Al-Mourabitoun                                          & 60 & {\color[HTML]{009901} GN} \\ \hline
< Houthis                                       & 13 & {\color[HTML]{FFCCC9} PK} & Revolutionary Organization 17 November                  & 61 & BK                        \\ \hline
< Al-Nusra Front                                & 14 & {\color[HTML]{FFCCC9} PK} & Holy Land Foundation for Relief and Development         & 62 & {\color[HTML]{F56B00} OR} \\ \hline
< Boko Haram                                    & 15 & {\color[HTML]{009901} GN} & Ansar al-Sharia (Libya)                                 & 63 & {\color[HTML]{009901} GN} \\ \hline
< Ulster Volunteer Force                        & 16 & BK                        & Al-Itihaad al-Islamiya                                  & 64 & {\color[HTML]{009901} GN} \\ \hline
< Shining Path                                  & 17 & BK                        & Al-Haramain Foundation                                  & 65 & {\color[HTML]{3531FF} BL} \\ \hline
< Popular Front for the Liberation of Palestine & 18 & {\color[HTML]{F56B00} OR} & Ansar Bait al-Maqdis                                    & 66 & {\color[HTML]{FFCCC9} PK} \\ \hline
< Lashkar-e-Taiba                               & 19 & {\color[HTML]{FE0000} RD} & Ansaru                                                  & 67 & {\color[HTML]{009901} GN} \\ \hline
< Hizb ut-Tahrir                                & 20 & {\color[HTML]{3531FF} BL} & Babbar Khalsa                                           & 68 & {\color[HTML]{3531FF} BL} \\ \hline
< Al-Qaeda in the Arabian Peninsula             & 21 & {\color[HTML]{FFCCC9} PK} & Jamaat-ul-Mujahideen Bangladesh                         & 69 & {\color[HTML]{FE0000} RD} \\ \hline
< Tehrik-i-Taliban Pakistan                     & 22 & {\color[HTML]{FE0000} RD} & Force 17                                                & 70 & {\color[HTML]{F56B00} OR} \\ \hline
< Islamic Jihad Mov. in Palestine           & 23 & {\color[HTML]{F56B00} OR} & Kata'ib Hezbollah                                       & 71 & {\color[HTML]{FFCCC9} PK} \\ \hline
< %Islamic Jihad Movement in Palestine           & KG23 & {\color[HTML]{F56B00} OR} & Kata'ib Hezbollah                                       & KG71 & {\color[HTML]{FFCCC9} PK} \\ \hline
< Ulster Defence Association                    & 24 & BK                        & Kurdistan Freedom Hawks                                 & 72 & BK                        \\ \hline
< Abu Sayyaf                                    & 25 & {\color[HTML]{FE0000} RD} & Islamic Jihad Union                                     & 73 & {\color[HTML]{FE0000} RD} \\ \hline
< Real Irish Republican Army                    & 26 & BK                        & Abdullah Azzam Brigades                                 & 74 & {\color[HTML]{FFCCC9} PK} \\ \hline
< Ansar Dine                                    & 27 & {\color[HTML]{009901} GN} & Moroccan Islamic Comb. Group                        & 75 & {\color[HTML]{009901} GN} \\ \hline
< %Ansar Dine                                    & KG27 & {\color[HTML]{009901} GN} & Moroccan Islamic Combatant Group                        & KG75 & {\color[HTML]{009901} GN} \\ \hline
< Jemaah Islamiyah                              & 28 & {\color[HTML]{FE0000} RD} & Ansar al-Sharia (Tunisia)                               & 76 & {\color[HTML]{009901} GN} \\ \hline
< Al-Qaeda in the Islamic Maghreb               & 29 & {\color[HTML]{009901} GN} & Al-Qaeda, Indian Subcontinent                     & 77 & {\color[HTML]{FE0000} RD} \\ \hline
< %Al-Qaeda in the Islamic Maghreb               & KG29 & {\color[HTML]{009901} GN} & Al-Qaeda in the Indian Subcontinent                     & KG77 & {\color[HTML]{FE0000} RD} \\ \hline
< Egyptian Islamic Jihad                        & 30 & {\color[HTML]{FFCCC9} PK} & Jund al-Aqsa                                            & 78 & {\color[HTML]{FFCCC9} PK} \\ \hline
< Al-Jama'a al-Islamiyya                        & 31 & {\color[HTML]{FFCCC9} PK} & Hezbollah Al-Hejaz                                      & 79 & {\color[HTML]{FFCCC9} PK} \\ \hline
< Jaish-e-Mohammed                              & 32 & {\color[HTML]{FE0000} RD} & Jamaat-ul-Ahrar                                         & 80 & {\color[HTML]{FE0000} RD} \\ \hline
< Aum Shinrikyo                                 & 33 & {\color[HTML]{FE0000} RD} & Jamaah Ansharut Tauhid                                  & 81 & {\color[HTML]{FE0000} RD} \\ \hline
< United Self-Defense Forces of Colombia        & 34 & BK                        & Islamic State of Iraq and the Levant ??? Algeria Province & 82 & {\color[HTML]{009901} GN} \\ \hline
< Armed Islamic Group of Algeria                & 35 & {\color[HTML]{009901} GN} & Osbat al-Ansar                                          & 83 & {\color[HTML]{FFCCC9} PK} \\ \hline
< Continuity Irish Republican Army              & 36 & BK                        & International Sikh Youth Federation                     & 84 & {\color[HTML]{FE0000} RD} \\ \hline
< Movement for Oneness and Jihad in West Africa & 37 & {\color[HTML]{009901} GN} & East Turkestan Liberation Organization                  & 85 & {\color[HTML]{FE0000} RD} \\ \hline
< Quds Force                                    & 38 & {\color[HTML]{FFCCC9} PK} & Great Eastern Islamic Raiders' Front                    & 86 & BK                        \\ \hline
< Al-Aqsa Martyrs' Brigades                     & 39 & {\color[HTML]{F56B00} OR} & Aden-Abyan Islamic Army                                 & 87 & {\color[HTML]{FFCCC9} PK} \\ \hline
< Com. Party of the Philippines            & 40 & {\color[HTML]{FE0000} RD} & Al-Aqsa Foundation                                      & 88 & {\color[HTML]{F56B00} OR} \\ \hline
< %Communist Party of the Philippines            & KG40 & {\color[HTML]{FE0000} RD} & Al-Aqsa Foundation                                      & KG88 & {\color[HTML]{F56B00} OR} \\ \hline
< Caucasus Emirate                              & 41 & {\color[HTML]{FE0000} RD} & Khalistan Zindabad Force                                & 89 & {\color[HTML]{FE0000} RD} \\ \hline
< Haqqani network                               & 42 & {\color[HTML]{FE0000} RD} & Mujahidin Indonesia Timur                               & 90 & {\color[HTML]{FE0000} RD} \\ \hline
< Turkistan Islamic Party                       & 43 & {\color[HTML]{FE0000} RD} & Al-Badr                                                 & 91 & {\color[HTML]{FE0000} RD} \\ \hline
< Ansar al-Islam                                & 44 & {\color[HTML]{FFCCC9} PK} & Soldiers of Egypt                                       & 92 & {\color[HTML]{FFCCC9} PK} \\ \hline
< Izz ad-Din al-Qassam Brigades                 & 45 & {\color[HTML]{F56B00} OR} & National Liberation Army                                & 93 & BK                        \\ \hline
< Lashkar-e-Jhangvi                             & 46 & {\color[HTML]{FE0000} RD} & Jundallah                                               & 94 & {\color[HTML]{FE0000} RD} \\ \hline
< Harkat-ul-Jihad al-Islami                     & 47 & {\color[HTML]{FE0000} RD} & Army of Islam                                           & 95 & {\color[HTML]{FFCCC9} PK} \\ \hline
< Islamic Movement of Uzbekistan                & 48 & {\color[HTML]{FE0000} RD} &                                                         &      &                           \\ \hline
< \end{tabular}
< \end{table}
< 
< 
< 
< \begin{table}[]
< \centering
< \caption{List of selected countries.}
< \label{tab:countries}
< \begin{tabular}{|c|c|c|c|c|c|}
< \hline
< Rank & Name           & abr & Rank & Name                 & abr \\ \hline
< 1    & United States  & US  & 33   & Portugal             & PT  \\ \hline
< 2    & France         & FR  & 34   & Ukraine              & UA  \\ \hline
< 3    & Germany        & DE  & 35   & Czech Republic       & CZ  \\ \hline
< 4    & United Kingdom & GB  & 36   & Malaysia             & MY  \\ \hline
< 5    & Iran           & IR  & 37   & Thailand             & TH  \\ \hline
< 6    & India          & IN  & 38   & Vietnam              & VN  \\ \hline
< 7    & Canada         & CA  & 39   & Nigeria              & NG  \\ \hline
< 8    & Australia      & AU  & 40   & Afghanistan          & AF  \\ \hline
< 9    & China          & CN  & 41   & Iraq                 & IQ  \\ \hline
< 10   & Italy          & IT  & 42   & Bangladesh           & BD  \\ \hline
< 11   & Japan          & JP  & 43   & Syria                & SY  \\ \hline
< 12   & Russia         & RU  & 44   & Morocco              & MA  \\ \hline
< 13   & Spain          & ES  & 45   & Algeria              & DZ  \\ \hline
< 14   & Netherlands    & NL  & 46   & Saudi Arabia         & SA  \\ \hline
< 15   & Poland         & PL  & 47   & Lebanon              & LB  \\ \hline
< 16   & Sweden         & SE  & 48   & Kazakhstan           & KZ  \\ \hline
< 17   & Mexico         & MX  & 49   & Albania              & AL  \\ \hline
< 18   & Turkey         & TR  & 50   & United Arab Emirates & AE  \\ \hline
< 19   & South Africa   & ZA  & 51   & Yemen                & YE  \\ \hline
< 20   & Switzerland    & CH  & 52   & Tunisia              & TN  \\ \hline
< 21   & Philippines    & PH  & 53   & Jordan               & JO  \\ \hline
< 22   & Austria        & AT  & 54   & Libya                & LY  \\ \hline
< 23   & Belgium        & BE  & 55   & Uzbekistan           & UZ  \\ \hline
< 24   & Pakistan       & PK  & 56   & Kuwait               & KW  \\ \hline
< 25   & Indonesia      & ID  & 57   & Qatar                & QA  \\ \hline
< 26   & Greece         & GR  & 58   & Mali                 & ML  \\ \hline
< 27   & Denmark        & DK  & 59   & Kyrgyzstan           & KG  \\ \hline
< 28   & South Korea    & KR  & 60   & Tajikistan           & TJ  \\ \hline
< 29   & Israel         & IL  & 61   & Oman                 & OM  \\ \hline
< 30   & Hungary        & HU  & 62   & Turkmenistan         & TM  \\ \hline
< 31   & Finland        & FI  & 63   & Chad                 & TD  \\ \hline
< 32   & Egypt          & EG  & 64   & South Sudan          & SS  \\ \hline
< \end{tabular}
< \end{table}
< 
< The matrix elements of $\GR, \Grr, \Gqr$ corresponding to the part of 
< 95 terrorist groups, are shown by color in Fig.~\ref{fig1} (the indices are ordered by 
< KG values from Table~\ref{tab:groups} so that an element with KG1=KG1 is in the top left corner).
< The largest matrix elements of $\GR$ correspond to top PageRank groups of Table~\ref{tab:groups}
< being dominated by the PageRank vector in $\Gpr$. The elements of $\Grr$ and $\Gqr$ are smaller
< but they determine direct and indirect interactions between groups.
< 
< 
< \begin{figure}[b]
< %\sidecaption
< \includegraphics[scale=0.195]{fig1}
< \caption{Density plots of matrices $\GR$, $\Grr$ and $\Gqrnd$ 
< (left, middle  and right; color changes from red at maximum to blue at zero); only the part of the 95 nodes of 
< Table~\ref{tab:groups} is shown.}
< \label{fig1}       % Give a unique label
< \end{figure}
< 
< According to  Fig.~\ref{fig1} the strong interactions between groups can be found by analyzing $\Gqr$ 
< providing new links absent in  $\Grr$. As an example we list:
< \textit{
< Tehrik-i-Taliban Pakistan (KG22) and Jundallah (KG94); 
< Hamas (KG5) and Izz ad-Din al-Qassam Brigades (KG45);
< Taliban (KG3) and Al-Qaeda in the Arabian Peninsula (KG21);
<  Kurdistan Freedom Hawks (KG72) and Kurdistan Workers' Party (KG9)}. 
< 
< \begin{figure}[b]
< %\sidecaption
< \includegraphics[scale=0.2]{fig2}
< \caption{Friendship network structure between terrorist groups obtained from $\Gqr$+$\Grr$; color marks 
< categories with top nodes
< (see text and  Table~\ref{tab:groups}), point size if proportional to PageRank probability of nodes;
< bold black arrows point to top 4 friends, gray tiny arrows show friends of friends interactions 
< computed until no new edges are added to the graph (drawn with \cite{gephi,hu}.}
< \label{fig2}       % Give a unique label
< \end{figure}
< 
< \begin{figure}[b]
< %\sidecaption
< \includegraphics[scale=0.42]{fig3}
< \caption{Right panel: friendship network structure extracted from $\Gqr+\Grr$ 
< with 6 top groups of each category (marked by its color) and
< 4 top friend countries for each group.
< Left panel:  the network in case of 2 friends for top groups of each category
< and top friend 2 countries for each group; countries are marked by cyan color for a better visibility
< (drawn with \cite{gephi,hu}.}
< \label{fig3}       % Give a unique label
< \end{figure}
< 
< To analyze the network structure of groups we attribute them to 6 different categories marked by 6 colors
< in Table~\ref{tab:groups}:  
< C1 of the international category for groups operating world wide (BL, top group KG1 ISIS),  
< C2 of the Asian category for Asian countries (RD, top group KG3 Taliban), 
< C3 of the conflict between Israel and Arab countries (OR, top group KG5 Hamas),
< C4 related to  African countries (GN, top group KG10 Al-Shabaab),
< C5 related to Middle East and Africa (PK, top group KG13 Houthis),
< C6 includes all remaining groups (BK, top group KG4 IRA).
< 
< For the analysis of network structure of groups we select top group nodes of each category 
< in Table~\ref{tab:groups} and then 
< top 4 friends for these group nodes 
< (with 4 largest matrix elements of $\Grr+\Gqrnd$) 
< inside a column of a given group corresponding
< to 4 largest outgoing link weight); then we consider friends of friends. The obtained network structure
< of groups is shown in Fig~\ref{fig2}. This structure clearly highlights the clustering of nodes
< corresponding to selected categories. It shows the leading role of top PageRank nodes of each category
< appearing on intersections of multiple links. The appearance of links due to indirect relations between 
< groups  is confirmed by the known facts. Thus  
< Saudi Arabia is an important funding source for Al-Shabaab \cite{alshabab1} 
< when Al-Qaeda in the Arabian Peninsula is the linking group between Al-Shabaab and Houthis 
< due to the fact that Houthis are facing Saudi Arabia.
< Hezbollah and Houthis share the same ideology, since they 
< are both Shiite and they are linked to Iran. Hamas and Hezbollah 
< share the same ideology of facing Israel and the results show that Hezbollah 
< is the linking group between Hamas and Houthis.
< As shown in Fig.~\ref{fig2}, the groups that are listed as International 
< are clearly playing that role by having lot of ingoing links from the other categories.
< 
< The interactions between groups and countries are characterized by the network structure
< shown in Fig.~\ref{fig3}. For clarity we first show only 4 country friends for 6 top 
< PageRank groups of each of 6 categories (right panel) and then we show
< more detailed structure with 2 group friends and 2 country friends (left panel). 
< The network reflects directly the real existing relations between groups and countries.
< Thus  Taliban is an active group in Afghanistan and Pakistan that represents
<  an Islamist militant organization that was one of the prominent factions 
< in the Afghan Civil War \cite{taliban1,taliban2,taliban3}. 
< As shown in Fig.\ref{fig3}, Afghanistan and Pakistan are the most influenced countries by Taliban.
< The fact that Saudi Arabia links Houthis, Taliban and Al Shabaab can be explained 
< by the fact that Saudi Arabia is in war with Houthis \cite{yemen1,yemen2}. 
< Also, the main fund of the groups that are active in Afghanistan and Pakistan 
< are from Saudi Arabia \cite{sauditaliban}. Moreover, Al-Shabab advocates 
< the Saudi-inspired Wahhabi version of Islam \cite{alshabab2}.
< Referring to \cite{isis}, ISIS was born in 2006 in Iraq as Islamic State of Iraq (ISI). 
< Its main activities are in Syria and Iraq. As shown in Fig.~\ref{fig3} 
< a strong relationship exists among the two countries and ISIS.
< Hamas and Hezbollah are the leading groups in MEA facing Israel. 
< As shown in Fig.\ref{fig3} and knowing the relationship between Hezbollah and Houthis, 
< we can explain why Israel is a linking node between Houthis and Hamas.
< Finally, we find that Iran links Houthis with ISIS. 
< This could be explained by the fact that both groups are in conflict with Saudi Arabia.
< 
< To analyze in a more detailed way the influence of specific terrorist groups on
< the selected 64 world countries we introduce the sensitivity $F$
< determined by the logarithmic derivatives of PageRank probability $P$ 
< obtained from $\GR$. At first we define $\delta$ as the relative fraction to be added to the relationship from node $j$ to node $i$ in $\GR$. 
< Knowing $\delta$, a new modified matrix $\GRprime$ is calculated in two steps. First, element $\GRprime(i,j)$ is set to $(1+\delta)\cdot\GR(i,j)$. Second, all elements of column $j$ of $\GRprime$ are normalized to 1 (including element $i$) to preserve the column-normalized property of this Perron-Frobenius matrix.
< $\GRprime$ now reflects an increased probability for going from node $j$ to node $i$.
< 
< It is now possible to calculate the modified PageRank eigenvector $\Pprime$ from $\GRprime$ using the standard $\GRprime\Pprime = \Pprime$ relation and compare it to the original PageRank probabilities $P$ calculated with $\GR$ using $\GR P = P$.
< Due to the relative change of the transition probability between nodes $i$ and $j$, steady state PageRank 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 probabilities:
< 
< \begin{equation}
< D_{(j \rightarrow i)} = \frac{{\rm d}P}{P} = \frac{(P-\Pprime)/\delta}{P} 
< \label{eq_sensitivity}
< \end{equation}
< We note that this approach is near to the sensitivity of the world trade
< in respect to price of specific product (e.g. gas or petroleum) 
< used in \cite{ermannwtn}.
< 
< \begin{figure}[b]
< %\sidecaption
< \includegraphics[scale=0.1]{fig4}
< %
< % If no graphics program available, insert a blank space i.e. use
< %\picplace{5cm}{2cm} % Give the correct figure height and width in cm
< %
< \caption{Wold map of influence terrorist groups on countries expressed by
< sensitivity $D_{(j \rightarrow i)}$ ($j$ is country index, $i$ is groups index, see text). Left column:
< Taliban KG3, Hamas KG5,  Houthis KG13 (top to bottom). Right column:
< ISIS KG1, Al Shabab KG10, IRA KG4 (top to bottom). Color bar 
< marks  $D_{(j \rightarrow i)}$ values with red for maximum and green for minimum influence.}
< \label{fig4}       % Give a unique label
< \end{figure}
< 
< 
< Fig.~\ref{fig4} shows the sensitivity influence $D$ of top 6 groups 
< of each category on all 64 countries. Here we see that Taliban KG3 
< has important influence on Afghanistan,  Pakistan, and Saudi Arabia
< and less influence on other countries. In contrast ISIS KG1 
< has a strong world wide influence with the main effects on
< Canada, Libya, USA, Saudi Arabia.
< The world maps show that the groups of the left column (Taliban, Hamas, Houthis)
< produce mainly local influence in the world.
< In contrast, the groups of the right column (ISIS, Al Shabab, IRA)
< spread their influence world wide. The presented results determine 
< the geopolitical influence of each terrorist group.
< 
< 
< Fig.~\ref{fig5} shows the influence (expressed by $D_{(j \rightarrow i)}$) 
< of a relation between one selected country $j$ 
< and one selected terrorist group $i$ on the other countries. 
< The results are shown for two countries being US (left panel) and 
< Saudi Arabia (right panel). The results show the enormous influence of Saudi Arabia
< on terrorist groups and other countries (almost all panel is in red). 
< The influence of USA is more selective.
< 
< 
< \begin{figure}[b]
< %\sidecaption
< \includegraphics[scale=0.31]{fig5}
< %
< % If no graphics program available, insert a blank space i.e. use
< %\picplace{5cm}{2cm} % Give the correct figure height and width in cm
< %
< \caption{World map of sensitivity influence $D_{(j \rightarrow i)}$ 
< of the relation between a selected country $j$ with a terrorist group $i$
< (represented by group index $i$ from Table~\ref{tab:groups} in vertical axis) 
< on a world country $k$ (represented by country index $j$ from Table~\ref{tab:countries}
< in horizontal axis $j \neq k$ is excluded) for two $k$ values: USA (left), 
< Saudi Arabia (right). Color shows
< $D_{(j \rightarrow i)}$ value is changing in the range $(-2.8 \cdot 10^{-4},2.1 \cdot 10^{-4})$) for USA
< and  $(-4.8 \cdot 10^{-3}, 10^{-3})$) for SA; minimum/maximum values 
< correspond to blue/red.}
< \label{fig5}       % Give a unique label
< \end{figure}
< 
< 
< \section{Discussion}
< \label{sec:4}
< 
< We apply the reduced Google matrix analysis 
< (Fig.~\ref{fig1}) to the network of articles of English Wikipedia
< for analysis of network structure of 94 terrorist groups and their influence on 64 world countries
< (159 selected articles).
< This approach takes into account all human knowledge  accumulated in Wikipedia 
< taking into account all indirect interactions between 159 selected articles and the huge
< information contained by   5416537 articles of Wikipedia with 122232932 links.
< The network structure obtained for the terrorist groups (Figs.~\ref{fig2},~\ref{fig3})
< clearly show the presence of 6 types (categories)
< of groups. The main groups in each category are determined from their PageRank.
< We show that the indirect or hidden links between terrorist groups and countries
< play an important role and are, in many cases, predominant over direct links. 
< The geopolitical influence of specific terrorist groups on world countries 
< is determined via the sensitivity of PageRank variation in respect to specific
< links between between groups and countries (Fig.~\ref{fig4}).
< We see the presence of terrorist groups with localized
< geographical influence (e.g. Taliban) and others with world wide influence (ISIS).
< The influence of selected countries on terrorist groups and other countries
< is also determined by the developed approach (Fig.6).
< The obtained results, tested on the publicly available data of Wikipedia,
< show the efficiency of the analysis.
< We argue that the reduced Google matrix approach can find further important
< applications for terror networks analysis using the more advanced and detailed data bases.
< 
< We thank Sabastiano Vigna \cite{vigna} for providing us his computer codes
< which we used for a generation of the English Wikipedia network (2017).
< These codes had been 
< developed in the frame of EC FET Open NADINE project (2012-2015) \cite{nadine}
< and used for Wikipedia (2013) data in \cite{eomplos}.
< This work was granted access to the HPC resources of 
< CALMIP (Toulouse) under the allocation 2017-P0110. 
< 
<  \input{referenc}
<  \end{document}
---
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> \def\K#1{\textcolor{red}{[[K] #1]}}
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> 
> \begin{document}
> 
> \title*{Analysis of world terror networks from \\ the reduced Google matrix of Wikipedia}
> % Use \titlerunning{Short Title} for an abbreviated version of
> % your contribution title if the original one is too long
> \author{Samer El Zant,  Klaus M. Frahm, Katia Jaffr\`es-Runser, Dima L. Shepelyansky}
> % Use \authorrunning{Short Title} for an abbreviated version of
> % your contribution title if the original one is too long
> \institute{Samer El Zant \at Institut de Recherche en Informatique de Toulouse/INPT, Universit\'e de Toulouse, Toulouse, France, \email{samer.elzant@enseeiht.fr}
> \and Klaus M. Frahm \at Lab. de Phys. Theorique, CNRS, Univ. de Toulouse, CNRS, UPS, 
> Toulouse, France, \email{frahm@irsamc.ups-tlse.fr}
> \and Katia Jaffr\`es-Runser \at Institut de Recherche en Informatique de Toulouse/INPT, Universit\'e de Toulouse, Toulouse, France, \email{kjr@enseeiht.fr}
> \and Dima L. Shepelyansky \at Lab. de Phys. Theorique, CNRS, Univ. de Toulouse, CNRS, UPS, 
> Toulouse, France, \email{dima@irsamc.ups-tlse.fr}
> }
> %
> % Use the package "url.sty" to avoid
> % problems with special characters
> % used in your e-mail or web address
> %
> \maketitle
> 
> 
> \abstract{
> We apply the reduced Google matrix method for analysis 
> of the interactions between 95 terrorist groups
> and determining their relations and influence on 64 world 
> countries. This is done on the basis of the Google matrix of the English Wikipedia
> (2017) composed of 5~416~537 articles which accumulate a great part of global
> human knowledge. The reduced Google matrix takes into account the
> direct and hidden links between a selection of 159 nodes (articles)
> appearing due to all paths of a random surfer moving over the whole network.
> As a result we obtain the network structure of terrorist groups
> and their relations with selected countries. Using the sensitivity of PageRank
> to a weight variation of specific links we determine the geopolitical
> sensitivity and influence of specific terrorist groups
> on world countries. We argue that this approach can find useful application for
> more extensive and detailed data bases analysis.  
> }
> 
> \section{Introduction}
> \label{sec:1}
> 
> {\it ''A new type of terrorism threatens the world,
> driven by networks of fanatics determined to inflict 
> maximam civilian and economic damages on distant targets in pursuit of 
> their extremist goals''} \cite{sageman1}. The origins of this world wide phenomenon
> are under investigation in political, social and religious sciences
> (see e.g.  \cite{kepel1,kepel2,sageman1,sageman2} and Refs. therein).
> At the same time the number of terrorist groups  is growing in the world \cite{taliban1}
> reaching over 100 officially recognized groups acting in various
> countries of the world \cite{wikispgroups}. These numbers become quite large and 
> the mathematical analysis of multiple interactions between these groups
> and their relationships to world countries is getting of great timeliness.
> The first steps in this direction are reported in a few publications 
> (see e.g. \cite{hicks,latora}) showing that the network science methods 
> (see e.g. \cite{dorogovtsev})
> should be well adapted to such type of investigations. However, it is 
> difficult to obtain a clear network structure with all dependencies which
> are emerging from the surrounding world with all its complexity.
> 
> In this work we use the approach of the Google matrix $G$ and PageRank algorithm
> developed by Brin and Page for large scale WWW network analysis \cite{brin}.
> The mathematical and statistical properties of this approach for various networks 
> are described in \cite{langville,rmp2015}. The efficiency of these methods
> are demonstrated for Wikipedia and world trade networks in 
> \cite{eomplos,ermannwtn,lages}. For the analysis of the terror networks
> we use the reduced Google matrix approach developed recently 
> \cite{frahm,politwiki,geop}. This approach selects from a global large scale network
> a subset of nodes of interest and constructs the reduced Google matrix 
> $\GR$ for this subset including all indirect links connecting the subset nodes 
> via the global network. The analysis of political leaders 
> and world countries subsets of Wikipedia networks in various language editions
> demonstrated the efficiency of this analysis \cite{politwiki,geop}. Here, for the 
> English Wikipedia network (collected in May 2017), we target a subset
> of $N_g = 95$ terrorist ones referenced in Wikipedia articles of groups
> enlisted as terrorist groups for at least two countries in \cite{wikispgroups} (see Table~\ref{tab:groups}).
> In addition we select the group of $N_c=64$ related world countries
> given in Table~\ref{tab:countries}. This gives us the size of $\GR$ being $N_r=N_g+N_c=159$
> that is much smaller then the global Wikipedia network with
> $N=5~416~537$ nodes (articles) and $N_\ell = 122~232~932$ links generated by 
> quotation links from one article to another. The method of the reduced Google matrix
> and the obtained results for interactions 
> between terrorist groups and countries  are described in the next Sections.
> 
> 
> 
> \section{Reduced Google matrix}
> \label{sec:2}
> 
> It is convenient to describe the network of $N$ Wikipedia articles by the Google matrix $G$ constructed from 
> the adjacency matrix $A_{ij}$ with elements $1$ if article (node) $j$ 
> points to  article (node) $i$ and zero otherwise. 
> In this case, elements of the Google matrix take the standard form 
> $G_{ij} = \alpha S_{ij} + (1-\alpha) / N$ \cite{brin,langville,rmp2015},
> where $S$ is the matrix of Markov transitions with elements  $S_{ij}=A_{ij}/k_{out}(j)$, 
> $k_{out}(j)=\sum_{i=1}^{N}A_{ij}\neq0$ being the node $j$ out-degree
> (number of outgoing links) and with $S_{ij}=1/N$ if $j$ has no outgoing links (dangling node). 
> Here $0< \alpha <1$ is the damping factor  which for a random surfer
> determines the probability $(1-\alpha)$ to jump to any node; below we use the standard value $\alpha=0.85$. 
> The right eigenvector
> of $G$ with the unit eigenvalue gives the PageRank probabilities
> $P(j)$ to find a random surfer on a node $j$. We order all nodes by decreasing probability $P$ 
> getting them ordered by the PageRank index $K=1,2,...N$ with a maximal probability at $K=1$.
> From this global ranking we obtain the local ranking of groups and countries given in 
> Tables~\ref{tab:groups},~\ref{tab:countries}.
> 
> 
> The reduced Google matrix $\GR$ is constructed for a selected subset of 
> nodes (articles) following the method described
> in \cite{frahm,politwiki} 
> %%% some proposition from Klaus:
> and based on concepts of scattering theory 
> used in different fields of mesoscopic and nuclear physics or 
> quantum chaos. 
> %%% maybe some additional citations behind ``chaos'' in the 
> %%% last phrase.
> This matrix has $N_r$ nodes and belongs 
> to the class of Google matrices. In addition the 
> PageRank probabilities of selected $N_r$ nodes are the same 
> as for the global network with $N$ nodes,
> up to a constant multiplicative factor taking into account that 
> the sum of PageRank probabilities over $N_r$
> nodes is unity. The matrix $\GR$ is represented as a sum of three matrices 
> (components)
> $\GR = \Grr + \Gpr + \Gqr$ \cite{politwiki}. 
> The first term $\Grr$ is given by the direct links between selected 
> $N_r$ nodes in the global $G$ matrix with $N$ nodes, 
> the second term $\Gpr$ is rather close to 
> the matrix in which each column is given by 
> the PageRank vector $P_r$, ensuring that PageRank probabilities of $\GR$ are 
> the same as for $G$ (up to a constant multiplier).
> Therefore  $\Gpr$ doesn't provide much information about direct 
> and indirect links between selected nodes.
> The most interesting is the third matrix $\Gqr$ which takes 
> into account all indirect links between
> selected nodes appearing due to multiple links via 
> the global network nodes $N$ \cite{frahm,politwiki}.
> The matrix  $\Gqr = \Gqrd + \Gqrnd$ has diagonal ($\Gqrd$)
> and nondiagonal ($\Gqrnd$) parts. The part $\Gqrnd$
> represents the main interest since it describes indirect interactions between nodes. 
> The explicit formulas as well as the mathematical and numerical computation 
> methods of all three components of $\GR$ are given 
> in \cite{frahm,politwiki,geop}. 
> 
> The selected groups and countries are given in Tables~\ref{tab:groups},~\ref{tab:countries}
> in order of their PageRank probabilities (given by KG rank column for groups and Rank column for countries, respectively). 
> All countries have PageRank probabilities being larger then those of terrorist groups so that they are well separated.
> 
> 
> 
> 
> % Please add the following required packages to your document preamble:
> % \usepackage[normalem]{ulem}
> % \useunder{\uline}{\ul}{}
> \begin{table}[]
> \centering
> \caption{List of selected terrorist groups (from \cite{wikispgroups}) attributed to 6 categories marked by color.}
> \label{tab:groups}
> %\begin{tabular}{|c|c|c|c|c|c|}
> \begin{tabular}{|p{4cm}|c|c|p{4cm}|c|c|}
> \hline
> Name                                          & KG  & Color                     & Name                                                    & KG  & Color                     \\ \hline
> Islamic State of Iraq and the Levant          & 1  & {\color[HTML]{3531FF} BL} & Hezb-e Islami Gulbuddin                                 & 49 & {\color[HTML]{FE0000} RD} \\ \hline
> Al-Qaeda                                      & 2  & {\color[HTML]{3531FF} BL} & Kach and Kahane Chai                                    & 50 & BK                        \\ \hline
> Taliban                                       & 3  & {\color[HTML]{FE0000} RD} & Palestine Liberation Front                              & 51 & {\color[HTML]{F56B00} OR} \\ \hline
> Provisional Irish Republican Army             & 4  & BK                        & Harkat-ul-Mujahideen                                    & 52 & {\color[HTML]{FE0000} RD} \\ \hline
> Hamas                                         & 5  & {\color[HTML]{F56B00} OR} & Kurdistan Free Life Party                               & 53 & BK                        \\ \hline
> Hezbollah                                     & 6  & {\color[HTML]{F56B00} OR} & Indian Mujahideen                                       & 54 & {\color[HTML]{FE0000} RD} \\ \hline
> Muslim Brotherhood                            & 7  & {\color[HTML]{3531FF} BL} & Abu Nidal Organization                                  & 55 & {\color[HTML]{F56B00} OR} \\ \hline
> Liberation Tigers of Tamil Eelam              & 8  & {\color[HTML]{FE0000} RD} & Hizbul Mujahideen                                       & 56 & {\color[HTML]{FE0000} RD} \\ \hline
> Kurdistan Workers' Party                      & 9  & BK                        & Libyan Islamic Fighting Group                           & 57 & {\color[HTML]{009901} GN} \\ \hline
> Al-Shabaab (militant group)                   & 10 & {\color[HTML]{009901} GN} & Islamic State of Iraq and the Levant in Libya           & 58 & {\color[HTML]{009901} GN} \\ \hline
> ETA (separatist group)                        & 11 & BK                        & Revolutionary People's Liberation Party/Front           & 59 & BK                        \\ \hline
> FARC                                          & 12 & BK                        & Al-Mourabitoun                                          & 60 & {\color[HTML]{009901} GN} \\ \hline
> Houthis                                       & 13 & {\color[HTML]{FFCCC9} PK} & Revolutionary Organization 17 November                  & 61 & BK                        \\ \hline
> Al-Nusra Front                                & 14 & {\color[HTML]{FFCCC9} PK} & Holy Land Foundation for Relief and Development         & 62 & {\color[HTML]{F56B00} OR} \\ \hline
> Boko Haram                                    & 15 & {\color[HTML]{009901} GN} & Ansar al-Sharia (Libya)                                 & 63 & {\color[HTML]{009901} GN} \\ \hline
> Ulster Volunteer Force                        & 16 & BK                        & Al-Itihaad al-Islamiya                                  & 64 & {\color[HTML]{009901} GN} \\ \hline
> Shining Path                                  & 17 & BK                        & Al-Haramain Foundation                                  & 65 & {\color[HTML]{3531FF} BL} \\ \hline
> Popular Front for the Liberation of Palestine & 18 & {\color[HTML]{F56B00} OR} & Ansar Bait al-Maqdis                                    & 66 & {\color[HTML]{FFCCC9} PK} \\ \hline
> Lashkar-e-Taiba                               & 19 & {\color[HTML]{FE0000} RD} & Ansaru                                                  & 67 & {\color[HTML]{009901} GN} \\ \hline
> Hizb ut-Tahrir                                & 20 & {\color[HTML]{3531FF} BL} & Babbar Khalsa                                           & 68 & {\color[HTML]{3531FF} BL} \\ \hline
> Al-Qaeda in the Arabian Peninsula             & 21 & {\color[HTML]{FFCCC9} PK} & Jamaat-ul-Mujahideen Bangladesh                         & 69 & {\color[HTML]{FE0000} RD} \\ \hline
> Tehrik-i-Taliban Pakistan                     & 22 & {\color[HTML]{FE0000} RD} & Force 17                                                & 70 & {\color[HTML]{F56B00} OR} \\ \hline
> Islamic Jihad Mov. in Palestine           & 23 & {\color[HTML]{F56B00} OR} & Kata'ib Hezbollah                                       & 71 & {\color[HTML]{FFCCC9} PK} \\ \hline
> %Islamic Jihad Movement in Palestine           & KG23 & {\color[HTML]{F56B00} OR} & Kata'ib Hezbollah                                       & KG71 & {\color[HTML]{FFCCC9} PK} \\ \hline
> Ulster Defence Association                    & 24 & BK                        & Kurdistan Freedom Hawks                                 & 72 & BK                        \\ \hline
> Abu Sayyaf                                    & 25 & {\color[HTML]{FE0000} RD} & Islamic Jihad Union                                     & 73 & {\color[HTML]{FE0000} RD} \\ \hline
> Real Irish Republican Army                    & 26 & BK                        & Abdullah Azzam Brigades                                 & 74 & {\color[HTML]{FFCCC9} PK} \\ \hline
> Ansar Dine                                    & 27 & {\color[HTML]{009901} GN} & Moroccan Islamic Comb. Group                        & 75 & {\color[HTML]{009901} GN} \\ \hline
> %Ansar Dine                                    & KG27 & {\color[HTML]{009901} GN} & Moroccan Islamic Combatant Group                        & KG75 & {\color[HTML]{009901} GN} \\ \hline
> Jemaah Islamiyah                              & 28 & {\color[HTML]{FE0000} RD} & Ansar al-Sharia (Tunisia)                               & 76 & {\color[HTML]{009901} GN} \\ \hline
> Al-Qaeda in the Islamic Maghreb               & 29 & {\color[HTML]{009901} GN} & Al-Qaeda, Indian Subcontinent                     & 77 & {\color[HTML]{FE0000} RD} \\ \hline
> %Al-Qaeda in the Islamic Maghreb               & KG29 & {\color[HTML]{009901} GN} & Al-Qaeda in the Indian Subcontinent                     & KG77 & {\color[HTML]{FE0000} RD} \\ \hline
> Egyptian Islamic Jihad                        & 30 & {\color[HTML]{FFCCC9} PK} & Jund al-Aqsa                                            & 78 & {\color[HTML]{FFCCC9} PK} \\ \hline
> Al-Jama'a al-Islamiyya                        & 31 & {\color[HTML]{FFCCC9} PK} & Hezbollah Al-Hejaz                                      & 79 & {\color[HTML]{FFCCC9} PK} \\ \hline
> Jaish-e-Mohammed                              & 32 & {\color[HTML]{FE0000} RD} & Jamaat-ul-Ahrar                                         & 80 & {\color[HTML]{FE0000} RD} \\ \hline
> Aum Shinrikyo                                 & 33 & {\color[HTML]{FE0000} RD} & Jamaah Ansharut Tauhid                                  & 81 & {\color[HTML]{FE0000} RD} \\ \hline
> United Self-Defense Forces of Colombia        & 34 & BK                        & Islamic State of Iraq and the Levant ??? Algeria Province & 82 & {\color[HTML]{009901} GN} \\ \hline
> Armed Islamic Group of Algeria                & 35 & {\color[HTML]{009901} GN} & Osbat al-Ansar                                          & 83 & {\color[HTML]{FFCCC9} PK} \\ \hline
> Continuity Irish Republican Army              & 36 & BK                        & International Sikh Youth Federation                     & 84 & {\color[HTML]{FE0000} RD} \\ \hline
> Movement for Oneness and Jihad in West Africa & 37 & {\color[HTML]{009901} GN} & East Turkestan Liberation Organization                  & 85 & {\color[HTML]{FE0000} RD} \\ \hline
> Quds Force                                    & 38 & {\color[HTML]{FFCCC9} PK} & Great Eastern Islamic Raiders' Front                    & 86 & BK                        \\ \hline
> Al-Aqsa Martyrs' Brigades                     & 39 & {\color[HTML]{F56B00} OR} & Aden-Abyan Islamic Army                                 & 87 & {\color[HTML]{FFCCC9} PK} \\ \hline
> Com. Party of the Philippines            & 40 & {\color[HTML]{FE0000} RD} & Al-Aqsa Foundation                                      & 88 & {\color[HTML]{F56B00} OR} \\ \hline
> %Communist Party of the Philippines            & KG40 & {\color[HTML]{FE0000} RD} & Al-Aqsa Foundation                                      & KG88 & {\color[HTML]{F56B00} OR} \\ \hline
> Caucasus Emirate                              & 41 & {\color[HTML]{FE0000} RD} & Khalistan Zindabad Force                                & 89 & {\color[HTML]{FE0000} RD} \\ \hline
> Haqqani network                               & 42 & {\color[HTML]{FE0000} RD} & Mujahidin Indonesia Timur                               & 90 & {\color[HTML]{FE0000} RD} \\ \hline
> Turkistan Islamic Party                       & 43 & {\color[HTML]{FE0000} RD} & Al-Badr                                                 & 91 & {\color[HTML]{FE0000} RD} \\ \hline
> Ansar al-Islam                                & 44 & {\color[HTML]{FFCCC9} PK} & Soldiers of Egypt                                       & 92 & {\color[HTML]{FFCCC9} PK} \\ \hline
> Izz ad-Din al-Qassam Brigades                 & 45 & {\color[HTML]{F56B00} OR} & National Liberation Army                                & 93 & BK                        \\ \hline
> Lashkar-e-Jhangvi                             & 46 & {\color[HTML]{FE0000} RD} & Jundallah                                               & 94 & {\color[HTML]{FE0000} RD} \\ \hline
> Harkat-ul-Jihad al-Islami                     & 47 & {\color[HTML]{FE0000} RD} & Army of Islam                                           & 95 & {\color[HTML]{FFCCC9} PK} \\ \hline
> Islamic Movement of Uzbekistan                & 48 & {\color[HTML]{FE0000} RD} &                                                         &      &                           \\ \hline
> \end{tabular}
> \end{table}
> 
> \begin{table}[]
> \centering
> \caption{List of selected countries.}
> \label{tab:countries}
> \begin{tabular}{|c|c|c|c|c|c|}
> \hline
> Rank & Name           & abr & Rank & Name                 & abr \\ \hline
> 1    & United States  & US  & 33   & Portugal             & PT  \\ \hline
> 2    & France         & FR  & 34   & Ukraine              & UA  \\ \hline
> 3    & Germany        & DE  & 35   & Czech Republic       & CZ  \\ \hline
> 4    & United Kingdom & GB  & 36   & Malaysia             & MY  \\ \hline
> 5    & Iran           & IR  & 37   & Thailand             & TH  \\ \hline
> 6    & India          & IN  & 38   & Vietnam              & VN  \\ \hline
> 7    & Canada         & CA  & 39   & Nigeria              & NG  \\ \hline
> 8    & Australia      & AU  & 40   & Afghanistan          & AF  \\ \hline
> 9    & China          & CN  & 41   & Iraq                 & IQ  \\ \hline
> 10   & Italy          & IT  & 42   & Bangladesh           & BD  \\ \hline
> 11   & Japan          & JP  & 43   & Syria                & SY  \\ \hline
> 12   & Russia         & RU  & 44   & Morocco              & MA  \\ \hline
> 13   & Spain          & ES  & 45   & Algeria              & DZ  \\ \hline
> 14   & Netherlands    & NL  & 46   & Saudi Arabia         & SA  \\ \hline
> 15   & Poland         & PL  & 47   & Lebanon              & LB  \\ \hline
> 16   & Sweden         & SE  & 48   & Kazakhstan           & KZ  \\ \hline
> 17   & Mexico         & MX  & 49   & Albania              & AL  \\ \hline
> 18   & Turkey         & TR  & 50   & United Arab Emirates & AE  \\ \hline
> 19   & South Africa   & ZA  & 51   & Yemen                & YE  \\ \hline
> 20   & Switzerland    & CH  & 52   & Tunisia              & TN  \\ \hline
> 21   & Philippines    & PH  & 53   & Jordan               & JO  \\ \hline
> 22   & Austria        & AT  & 54   & Libya                & LY  \\ \hline
> 23   & Belgium        & BE  & 55   & Uzbekistan           & UZ  \\ \hline
> 24   & Pakistan       & PK  & 56   & Kuwait               & KW  \\ \hline
> 25   & Indonesia      & ID  & 57   & Qatar                & QA  \\ \hline
> 26   & Greece         & GR  & 58   & Mali                 & ML  \\ \hline
> 27   & Denmark        & DK  & 59   & Kyrgyzstan           & KG  \\ \hline
> 28   & South Korea    & KR  & 60   & Tajikistan           & TJ  \\ \hline
> 29   & Israel         & IL  & 61   & Oman                 & OM  \\ \hline
> 30   & Hungary        & HU  & 62   & Turkmenistan         & TM  \\ \hline
> 31   & Finland        & FI  & 63   & Chad                 & TD  \\ \hline
> 32   & Egypt          & EG  & 64   & South Sudan          & SS  \\ \hline
> \end{tabular}
> \end{table}
> 
> \FloatBarrier
> 
> 
> 
> \section{Results}
> \label{sec:3}
> 
> 
> 
> In this work we extract from $\GR$ a network of 64 countries and 95 groups. 
> This network reflects direct and indirect interactions between countries and groups, 
> which motivates us to study the relative influence of group alliances 
> on the other ones and on the countries. 
> The matrix $\GR$ and its three components $\Grr$, $\Gpr$ and $\Gqr$ 
> are computed for $N_r=165$ Wikipedia network nodes 
> formed by $N_c=64$ country nodes and $N_g=95$ group nodes.  
> The weights of these three $\GR$ components are 
> $\Wrr$=0.0644, $\Wpr$=0.8769 and $\Wqr$=0.0587 (the weight is given by sum of all matrix elements divided by $N_r$,
> thus $\Wrr + \Wqr + \Wqr = 1$). 
> The dominant component is $\Gpr$ but as stated above it is approximately given by
> columns of PageRank vector so that the most interesting information is provided by
> $\Grr$ and especially the component $\Gqr$ given by indirect links \cite{politwiki,geop}.
> 
> 
> 
> 
> 
> The matrix elements of $\GR, \Grr, \Gqr$ corresponding to the part of 
> 95 terrorist groups are shown in the colormaps of Fig.~\ref{fig1} (indices are ordered by increasing 
> values of KG as given in Table~\ref{tab:groups}, thus element with KG1=KG1 is located at the top left corner).
> The largest matrix elements of $\GR$ are the ones of top PageRank groups of Table~\ref{tab:groups}. Such large values are enforced by $\Gpr$ component which is dominated by PageRank vector. 
> The elements of $\Grr$ and $\Gqr$ are smaller
> but they determine direct and indirect interactions between groups.
> 
> 
> 
> According to  Fig.~\ref{fig1} the strong interactions between groups can be found by analyzing $\Gqr$ 
> looking at new links appearing in $\Gqr$ and being absent from $\Grr$. As an example we list:
> \begin{itemize}
> \item[-] Tehrik-i-Taliban Pakistan (KG22) and Jundallah (KG94); 
> \item[-] Hamas (KG5) and Izz ad-Din al-Qassam Brigades (KG45);
> \item[-] Taliban (KG3) and Al-Qaeda in the Arabian Peninsula (KG21);
>  \item[-] Kurdistan Freedom Hawks (KG72) and Kurdistan Workers' Party (KG9). 
> \end{itemize}
> 
> \begin{figure}[h]
> %\sidecaption
> \includegraphics[scale=0.195]{fig1}
> \caption{Density plots of matrices $\GR$, $\Grr$ and $\Gqrnd$ 
> (left, middle  and right; color changes from red at maximum to blue at zero); only 95 terrorist nodes of 
> Table~\ref{tab:groups} are shown.}
> \label{fig1}       % Give a unique label
> \end{figure}
> 
> 
> \subsection{Network structure of groups}
> To analyze the network structure of groups we attribute them to 6 different categories marked by 6 colors
> in Table~\ref{tab:groups}:  C1 for the International category of groups operating worldwide (BL, top group is KG1 ISIS) ;  
> C2 for the groups targeting Asian countries (RD, top group is KG3 Taliban) ; C3 for the groups related with the Israelo-Arab conflict (OR, top group is KG5 Hamas) ; C4 for the groups targeting African countries (GN, top group is KG10 Al-Shabaab) ; C5 for the groups related to Middle Eastern and African countries (PK, top group is KG13 Houthis) ; C6 for all remaining groups (BK, top group is KG4 IRA).
> 
> We analyse the network structure of groups by selecting the top group node of each category 
> in Table~\ref{tab:groups} and then, their top 4 friends in $\Grr+\Gqrnd$ (i.e. the nodes with the 4 largest matrix elements of $\Grr+\Gqrnd$ in the column the group of interest. It corresponds to the 4 largest outgoing link weights). From the set of top group nodes and their top 4 friends, we continue to extract the top 4 friends of friends until no new node is added to this network of friends. The obtained network structure of groups is shown in Fig~\ref{fig2}. 
> This network structure clearly highlights the clustering of nodes corresponding to selected categories. It shows the leading role of top PageRank nodes for each category appearing as highly central nodes with large in-degree. 
> 
> \K{Samer, Peux tu verifier si ma reformulation est correcte dans ce paragraphe ?}
> The appearance of links due to indirect relationships between groups is confirmed by well-known facts. 
> For instance, it  can be seen that Al-Qaeda in the Arabian Peninsula (KG21) is linking Al-Shabaab (KG10) and Houthis (KG13). Al-Qaeda in the Arabian Peninsula is primarily active in Saudi Arabia. 
> It is well known that Saudi Arabia is an important financial support of Al-Shabaab \cite{alShabab1} and that Houthis is confronting Saudi Arabia. As such, it makes sense that Al-Qaeda in the Arabian Peninsula links both groups as it is tied to Saudi Arabia. 
> 
> \begin{figure}[h]
> %\sidecaption
> \includegraphics[scale=0.21]{fig2}
> \caption{Friendship network structure between terrorist groups obtained from $\Gqr$+$\Grr$; colors mark 
> categories of nodes and top nodes are given in text and  Table~\ref{tab:groups}; circle size is proportional to PageRank probability of nodes;
> bold black arrows point to top 4 friends, gray tiny arrows show friends of friends interactions 
> computed until no new edges are added to the graph (drawn with \cite{gephi,hu}.}
> \label{fig2}       % Give a unique label
> \end{figure}
> 
> Another meaningful example is the one of Hezbollah (KG6) and Houthis that share the same ideology, since they are both Shiite and are strongly linked to Iran. From Fig.~\ref{fig2}, it can be seen that Hezbollah is a direct friend of Houthis. The case of Hamas (KG5) and Hezbollah, that share the same ideology in facing Israel, is highlighted as well in our results. Moreover, Fig.~\ref{fig2} shows as well that Hezbollah is the linking group between Hamas and Houthis. Finally, the network of Fig.~\ref{fig2} clearly shows that the groups that are listed as International (blue color) are clearly playing that role by having lots of ingoing links from the other categories.
> 
> 
> 
> \subsection{Relationships between groups and countries}
> The interactions between groups and countries are characterized by the network structure
> shown in Fig.~\ref{fig3}. For clarity, we first show on the right panel of Fig.~\ref{fig3} the top 4 country friends of the 6 terrorist groups identified as leading each category. On the left panel, we show for the same 6 leading terrorist groups the top 2 country friends and top 2 terrorist groups friends. This latter representation shows altogether major ties between groups and countries and in-between groups. Very interesting and realistic relations between groups and countries can be extracted from this network. For instance, Taliban (KG3) is an active group in Afghanistan and Pakistan that represents an Islamist militant organization that was one of the prominent factions  in the Afghan Civil War \cite{taliban1,taliban2,taliban3}. As shown in Fig.\ref{fig3}, Afghanistan and Pakistan are the most influenced countries by Taliban.
> 
> 
> \begin{figure}[h]
> %\sidecaption
> \includegraphics[scale=0.43]{fig3}
> \caption{Right panel: friendship network structure extracted from $\Gqr+\Grr$ 
> with the top terrorist group of each category (marked by their respective colors) and
> their top 4 friend countries.
> Left panel:  the network in case of 2 friends for top groups of each category
> and top friend 2 countries for each group; countries are marked by cyan color for a better visibility
> (drawn with \cite{gephi,hu}.}
> \label{fig3}       % Give a unique label
> \end{figure}
> 
> The fact that Saudi Arabia links Houthis, Taliban and Al Shabaab can be explained 
> by the fact that Saudi Arabia is in war with Houthis \cite{yemen1,yemen2}. 
> Also, the main funding sources for groups active in Afghanistan and Pakistan 
> originate from Saudi Arabia \cite{sauditaliban}. Moreover, Al-Shabaab advocates 
> for the Saudi-inspired Wahhabi version of Islam \cite{alShabab2}.
> Referring to \cite{isis}, ISIS (KG1) was born in 2006 in Iraq as Islamic State of Iraq (ISI). 
> Its main activities are in Syria and Iraq. As shown in Fig.~\ref{fig3} 
> a strong relationship exists among the two countries and ISIS.
> Hamas and Hezbollah are the leading groups in MEA facing Israel. 
> As shown in Fig.\ref{fig3} and knowing the relationship between Hezbollah and Houthis, 
> we can explain why Israel is a linking node between Houthis and Hamas.
> Finally, we find that Iran links Houthis with ISIS. 
> This could be explained by the fact that both groups are in conflict with Saudi Arabia.
> 
> 
> \subsection{Sensitivity analysis}
> To analyze in a more detailed way the influence of specific terrorist groups on
> the selected 64 world countries we introduce the sensitivity $F$
> determined by the logarithmic derivatives of PageRank probability $P$ 
> obtained from $\GR$. At first we define $\delta$ as the relative fraction to be added to the relationship from node $j$ to node $i$ in $\GR$. 
> Knowing $\delta$, a new modified matrix $\GRprime$ is calculated in two steps. First, element $\GRprime(i,j)$ is set to $(1+\delta)\cdot\GR(i,j)$. Second, all elements of column $j$ of $\GRprime$ are normalized to 1 (including element $i$) to preserve the column-normalized property of this Perron-Frobenius matrix.
> $\GRprime$ now reflects an increased probability for going from node $j$ to node $i$.
> 
> It is now possible to calculate the modified PageRank eigenvector $\Pprime$ from $\GRprime$ using the standard $\GRprime\Pprime = \Pprime$ relation and compare it to the original PageRank probabilities $P$ calculated with $\GR$ using $\GR P = P$.
> Due to the relative change of the transition probability between nodes $i$ and $j$, steady state PageRank 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 probabilities:
> 
> \begin{equation}
> D_{(j \rightarrow i)} = \frac{{\rm d}P}{P} = \frac{(P-\Pprime)/\delta}{P} 
> \label{eq_sensitivity}
> \end{equation}
> $D_{(j \rightarrow i)}$ is a vector and $D_{(j \rightarrow i)}(k)$ gives for node $k$ it sensitivity to the change of link $j$ to $i$. 
> We note that this approach is similar to the sensitivity analysis of the world trade network to the price of specific products (e.g. gas or petroleum) as studied in \cite{ermannwtn}.
> 
> \begin{figure}[h]
> %\sidecaption
> \includegraphics[scale=0.1]{fig4}
> %
> % If no graphics program available, insert a blank space i.e. use
> %\picplace{5cm}{2cm} % Give the correct figure height and width in cm
> %
> \caption{Wold map of the influence of terrorist groups on countries expressed by
> sensitivity $D_{(j \rightarrow i)}(j)$ where $j$ is the country index and $i$ the group index, see text). Left column:
> Taliban KG3, Hamas KG5,  Houthis KG13 (top to bottom). Right column:
> ISIS KG1, Al Shabaab KG10, IRA KG4 (top to bottom). Color bar 
> marks  $D_{(j \rightarrow i)}(j)$ values with red for maximum and green for minimum influence.}
> \label{fig4}       % Give a unique label
> \end{figure}
> 
> 
> Fig.~\ref{fig4} shows maps of the sensitivity influence $D$ of the top groups 
> of the 6 categories on all 64 countries. Here we see that Taliban (KG3) 
> has important influence on Afghanistan,  Pakistan, and Saudi Arabia
> and less influence on other countries. In contrast ISIS (KG1) 
> has a strong worldwide influence with the main effects on
> Canada, Libya, USA, Saudi Arabia.
> The world maps show that the groups of the left column (Taliban, Hamas, Houthis)
> produce mainly local influence in the world.
> In contrast, the groups of the right column (ISIS, Al Shabaab, IRA)
> spread their influence worldwide. The presented results determine 
> the geopolitical influence of each terrorist group.
> 
> 
> Fig.~\ref{fig5} shows the influence of a relation between one selected country $c$ 
> and one selected terrorist group $i$ on the other countries $j$.  
> The results are shown for two countries being US (left panel - $c=1$) and 
> Saudi Arabia (right panel - $c=46$). Each element $(i,j)$ of the given matrices 
> is expressed by $D_{(c \rightarrow i)}(j)$). 
> Results show the enormous influence of Saudi Arabia
> on terrorist groups and other countries (almost all panel is in red). 
> The influence of USA is more selective.
> 
> 
> \begin{figure}[t]
> %\sidecaption
> \includegraphics[scale=0.32]{fig5}
> %
> % If no graphics program available, insert a blank space i.e. use
> %\picplace{5cm}{2cm} % Give the correct figure height and width in cm
> %
> \caption{World map of sensitivity influence $D_{(c \rightarrow i)}(j)$ 
> for the relation between a selected country $c$ and a terrorist group $i$
> (represented by group index $i$ from Table~\ref{tab:groups} in vertical axis) 
> on a world country $j$ (represented by country index $j$ from Table~\ref{tab:countries}
> in horizontal axis $j \neq k$ is excluded) for two $c$ values: USA (left), 
> Saudi Arabia (right). Color shows
> $D_{(c \rightarrow i)}(j)$ value is changing in the range $(-2.8 \cdot 10^{-4},2.1 \cdot 10^{-4})$) for USA
> and  $(-4.8 \cdot 10^{-3}, 10^{-3})$) for SA; minimum/maximum values 
> correspond to blue/red.}
> \label{fig5}       % Give a unique label
> \end{figure}
> 
> 
> \section{Discussion}
> \label{sec:4}
> 
> We have applied the reduced Google matrix analysis 
> (Fig.~\ref{fig1}) to the network of articles of English Wikipedia to analyse the
>  network structure of 94 terrorist groups and their influence over 64 world countries
> (159 selected articles).
> This approach takes into account all human knowledge  accumulated in Wikipedia, 
> leveraging all indirect interactions existing between the 159 selected articles and the huge
> information contained by  5~416~537 articles of Wikipedia and its 122~232~932 links.
> The network structure obtained for the terrorist groups (Figs.~\ref{fig2},~\ref{fig3})
> clearly show the presence of 6 types (categories)
> of groups. The main groups in each category are determined from their PageRank.
> We show that the indirect or hidden links between terrorist groups and countries
> play an important role and are, in many cases, predominant over direct links. 
> The geopolitical influence of specific terrorist groups on world countries 
> is determined via the sensitivity of PageRank variation in respect to specific
> links between groups and countries (Fig.~\ref{fig4}).
> We see the presence of terrorist groups with localized
> geographical influence (e.g. Taliban) and others with worldwide influence (ISIS).
> The influence of selected countries on terrorist groups and other countries
> is also determined by the developed approach (Fig.6).
> The obtained results, tested on the publicly available data of Wikipedia,
> show the efficiency of the analysis.
> We argue that the reduced Google matrix approach can find further important
> applications for terror networks analysis using more advanced and detailed databases.
> 
> \section{Acknowledgements}
> We thank Sabastiano Vigna \cite{vigna} for providing us his computer codes
> which we used for a generation of the English Wikipedia network (2017).
> These codes had been 
> developed in the frame of EC FET Open NADINE project (2012-2015) \cite{nadine}
> and used for Wikipedia (2013) data in \cite{eomplos}.
> This work was granted access to the HPC resources of 
> CALMIP (Toulouse) under the allocation 2017-P0110. 
> This work has been supported by the \emph{GOMOBILE} project supported jointly by 
> University of Toulouse APR 2015 and R?gion Occitanie under doctoral research grant \#15050459, 
> and in part by CHIST-ERA MACACO project, ANR-13-CHR2-0002-06.
> 
>  \input{referenc}
>  \end{document}
