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\jyear{2021}%

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\begin{document}

\title[Criminal network disruption]{Novel protocol for criminal network disruption}

%%=============================================================%%
%% Prefix	-> \pfx{Dr}
%% GivenName	-> \fnm{Joergen W.}
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%% \author*[1,2]{\pfx{Dr} \fnm{Joergen W.} \spfx{van der} \sur{Ploeg} \sfx{IV} \tanm{Poet Laureate} 
%%                 \dgr{MSc, PhD}}\email{iauthor@gmail.com}
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\author*[1,2]{\fnm{A.S.O.} \sur{Toledo}}\email{asotoledo@hotmail.com}

\author[3]{\fnm{Laura C.} \sur{Carpi}}\email{lauracarpi@gmail.com}
\equalcont{These authors contributed equally to this work.}

\author[1,4]{\fnm{A.P.F.} \sur{Atman}}\email{atman@cefetmg.br}
\equalcont{These authors contributed equally to this work.}

\author[1]{\fnm{A.P. Ba\^eta} \sur{Scarpelli}}\email{scarpelli@cefetmg.br}
\equalcont{These authors contributed equally to this work.}


\affil[1]{\orgdiv{Programa de P\'os-Gradua\c c\~ao em Modelagem Matem\'atica e Computacional}, \orgname{Centro Federal de Educa\c c\~ao Tecnol\'ogica de Minas Gerais}, \orgaddress{\country{Brazil}}}

\affil[2]{\orgname{Instituto Brasileiro de Seguran\c ca P\'ublica}, \orgaddress{\country{Brazil}}}

\affil[3]{\orgdiv{Escola de Engenharia}, \orgname{Universidade Federal de Minas Gerais}, \orgaddress{\country{Brazil}}}

\affil[4]{\orgname{Instituto Nacional de Ci\^encia e Tecnologia de Sistemas Complexos}, \orgaddress{\country{Brazil}}}



 \abstract{\textbf{Purpose:} Identify and characterize criminal organizations as a network of collaborative individuals, exposing their underlying structure as a multiplex network. Study different mechanisms to disrupt criminal networks and propose an effective protocol to assist the criminal investigation.
% 
 \textbf{Methods:} From data collected by security intelligence from the State of Minas Gerais, Brazil, we build crime networks using Graph Theory.  Different strategies to break the network are tested based on key agents identified by the node properties, organized by different crimes in a multilayer structure.
% 
 \textbf{Results:} The performance of nine different interruption strategies is quantified and analyzed, and a new protocol is proposed considering the greater efficiency compared to standard techniques.
% 
 \textbf{Conclusion:} A protocol to disrupt crime networks is proposed which can effectively assist police operations to be targeted to remove key criminals. The proposed method of disruption based on key agents allows police to identify, visualize and qualify criminal activity, increasing the chances of a successful criminal network disruption.}


\keywords{Criminal network, diversity, multiplex network, disruption, resilience}

%%\pacs[JEL Classification]{D8, H51}

%%\pacs[MSC Classification]{35A01, 65L10, 65L12, 65L20, 65L70}

\maketitle

\section{Introduction}\label{sec1}

Implementing new information technologies for police work is a great challenge to contain the growth of  crime \cite{egbert2021,wheeler2021}. The tools for this must be embedded in the institutional, organizational, and practical contexts of police work, allowing for policing based on socio-technical practices to disrupt crime and reduce the practice of crimes \cite{caplan2021}. Among the most challenging crimes, we find cyber-crime, child pornography, drug trafficking, and identity theft, which require proactive interventions to reach the criminal organizations that support them \cite{egbert2021}. Throughout the world, government agencies, scholars, and law enforcement are constantly seeking more effective ways to break or control these illegal structures. Technologies have been developed to study and simulate human relations, combining the real world and computing, to verify and create models representing how these relationships are established \cite{bright2021}. This makes it possible to standardize, with agility and accuracy, behaviors that can be identified and used as reference points in the search for ways to break the network of criminals \cite{duijn2014}. Since the 1990s, the network approach to studying criminal organizations has increased rapidly \cite{varese2013}. Network analysis techniques have been successfully used to understand better the interactions between criminals  \cite{bouchard2013,papachristos2017,toledo2020}.

Networked organizations are flexible and dynamic, capable of limiting exposure to risks, allowing alliances that depend on roles and the level of confidence among criminals \cite{campana2016}. These intrinsic adaptive characteristics make criminal networks challenging to identify and dismantle \cite{duijn2014}. Among the well-known complex network’s tools, centrality measures are usually used to analyze crime networks. If we consider the structure of social networks in general, individuals with essential roles display a higher degree of centrality than those whose roles are less significant \cite{bright2021}. In fact, this assumption is valid to social networks, which are primarily assortative networks \cite{catanzaro2004,croft2005,fisher2017}.  Criminal networks are disassortative networks, as will be shown in this work. In recent years, a novel and more realistic vision regarding modeling complex systems through networks has been progressing \cite{bright2021}. Complex systems are not isolated but inserted in an extensive network of networks has been gaining a lot of interest \cite{bargigli2015}. Multilayer networks appeared as a novel approach for these interconnected structures in which each participant network is represented as a layer. 

Multiplex networks are a particular case of multilayer networks, where layers are composed of the same group of nodes. Their interactions in the different layers occur only with their replica. A multiplex social network, for example, can be constructed considering the same group of individuals who participates in various social networks (Facebook\textsuperscript{\textcopyright}, Twitter\textsuperscript{\textcopyright}, Instagram\textsuperscript{\textcopyright}). Similarly, individuals involved in criminal activities usually experience different crimes, which can be represented in layers, which can be modeled as multiplex networks \cite{toledo2020,diviak2019,guo2016}.

A better understanding of these criminal networks is crucial for public security intelligence. Precise identification of individuals that play key roles in these organizations will allow for a more efficient interruption or prevention of criminal activities. This work aims to study a criminal network as a multiplex structure to efficiently analyze strategies to disrupt criminal networks.
%

\section{Criminal Data}
%

The complete database contains 152,973 records of individuals related to various crimes (Fig.~\ref{fig:fig1_networks}-a). Connections between criminals correspond to physical participation in crimes, being part of the same gang, phone calls interceptions, among others. This database contains 7,609 criminal networks, from 2 to 3,231 individuals per network.

\begin{figure}[ht]
\centering
\includegraphics[width=11cm]{fig1_networks.png}
\caption{[a] Occurrence link network containing 34,505 nodes (individuals) and 46,239 edges different types of connections present in the database. 
[b] The network used in our case study has 92 individuals (offenders and co-offenders) and 139 edges extracted from the database of this study. This network is used for measurements, analyzes, modeling, and simulations of this work.}
\label{fig:fig1_networks}
\end{figure}

This study uses a network of individuals who were the target of an intelligence operation in Belo Horizonte, State of Minas Gerais/Brazil, where several participants have been arrested. At that time, identifying a criminal network containing 58 individuals and 77 connections was possible after three months of hard work consuming human efforts and material resources. During the intelligence operation, individuals 51 and 80 have been identified as the leaders of this criminal network. The network considered for this study (Fig.~\ref{fig:fig1_networks}-b), Target Network, corresponds to a network containing 92 individuals and 139 links selected from a database collected through observations of intelligence agents of the Military Police of Minas Gerais.
%
\section{Complex networks tools}
%
In this study, four measures have been used to identify the different roles of the criminal network participants: degree centrality, betweenness centrality, multiplexity, and node diversity \cite{newman2010,sole2014,crossley2009,carpi2019}. 

Degree centrality (Equation \ref{eq:degree}) is the number of direct contacts the individual has, which is simply the number of direct connections of the individual. The higher the number of first neighbors, the higher the value of this measure. 

\begin{equation}
    C_{D}(i)= \frac{deg(i)}{n-1}  
    \label{eq:degree}
\end{equation}

Betweenness centrality (Equation \ref{eq:betweenness}) quantifies the number of times an individual appears as a bridge along the shortest path between two other individuals. The identification of roles in criminal networks mainly uses measures of centrality \cite{bright2021}.  

 \begin{equation}
    C_{B}(i)= \frac{1}{(n-1)(n-2)}    \sum_{j \neq i, k \neq i, j \neq k}{ \frac{ N_{sp}(j {\overset{i}{\longrightarrow}} k) }{ N_{sp} (j \longrightarrow k)} }   
    \label{eq:betweenness}
\end{equation}

Next, to study the node’s multiplexity and node diversity values, we construct a multiplex network where each layer corresponds to a specific crime. The crimes considered are ideological falsehood, arms trafficking, kidnapping, weapon carrying, homicide, theft, and drug trafficking. The multiplex network is illustrated in Fig.~\ref{fig:fig2_crimemultiplex}, showing the 77 individuals in the related crimes. Another 15 individuals of the Target Network are co-offenders and are not analyzed on the multiplex network.

\begin{figure}[ht!]
\centering
\includegraphics[width=11cm]{fig2_crime_multiplex.png}
\caption{Multiplex criminal networks represented in the structure are ideological falsehood (2 edges/4 nodes), arms trafficking (3 edges/4 nodes), kidnapping (13 edges/7 nodes), weapon carrying (35 edges/27 nodes), homicide (31 edges/29 nodes), theft (60 edges/45 nodes) and drug trafficking (79 edges/59 nodes); a total of 232 links and 77 individuals.}
\label{fig:fig2_crimemultiplex}
\end{figure}

Multiplexity is a type of representation that allows identifying strong relationships by looking at the repetition of links between nodes in the different layers \cite{de2013}, and link redundancy can consequently be associated with the level of trust or between individuals. Multiplexity is the number $N$ of layers the node participates in divided by the network's total number of layers $C$.

Finally, the diversity measure quantifies the heterogeneity of a node's connectivity patterns in the different layers \cite{carpi2019}. Diversity is a very appealing measure to study criminal networks since it can be associated with network resilience. A node with a high diversity value can reach more nodes, increasing the possibility of easy network adaptations in changes. We use a node diversity measure introduced for multiplex structures and based on the distances between its elements (nodes or layers) \cite{carpi2019}, given by Equation \ref{eq:formulaDI}. It is computed at a local level, which refers to the diversity of the connections of a node in the various layers. At a global level, it relates to the diversity of connectivity patterns of the whole system. A high diversity value indicates an increased number of different link configurations. A low diversity value indicates a common variety of link configurations and the presence of redundant links. This measure was successfully applied to study several systems, such as crime, biological, and transportation networks \cite{toledo2020,carpi2019,oliveira2020}. Criminal networks can absorb and resist disturbances, adapting to changes when necessary. This capacity is called network resilience. It has been shown in \cite{duijn2014} that the more diverse a structure is in terms of connections, the more resilient it becomes. This happens due to more options to compensate for changes in human capital and find new ways to continue operating efficiently.

\begin{equation}
  {\mathcal D_i} (\overline{p},\overline{q})=\frac{\sqrt{{\cal J} (\mathcal{N}^{\overline{p}}_i,\mathcal{N}^{\overline{q}}_i)} + \sqrt{{\cal J}  (T^{\overline{p}}_{i},T^{\overline{q}}_{i})}}{2 \sqrt{\log{(2)}}},    
 \label{eq:formulaDI}
\end{equation}

An important measure to characterize social networks is coefficient assortativity. Coefficient assortativity is a measure of correlation that shows whether nodes of high degree connect to nodes of the same or similar degree (assortative) or whether nodes of high degree connect to nodes of different degrees (disassortative) \cite{newman2010}. The assortativity coefficient is given by Equation \ref{eq:assortative}.

\begin{equation}
	r= \frac{\sum_{(i,j) \in E}{(f(i)-   \overline{f}_{1})(f(j)-   \overline{f}_{2}) } }{ \sqrt{\sum_{(i,j) \in E}{(f(i)-   \overline{f}_{1})^2}}  \sqrt{\sum_{(i,j) \in E}{(f(j)-   \overline{f}_{2})^2}} }   
	\label{eq:assortative}
\end{equation}
%
\section{Results}
%
The assortativity coefficient of 56 social networks was measured in this work, using datasets \cite{rossi2015}. As a result, 32 of these networks were found to be assortative, supporting \cite{fisher2017}, \textit{“Social networks are typically considered distinct from other networks because they are assortative”}.

The assortativity coefficient of the Target Network was measured at -0.233, indicating that it is a disassortative network. Measurements were performed on a sample of 239 networks, with more than 30 individuals, from criminal database networks, and all results were a negative coefficient. The disassortative characteristic of crime networks suggests that individuals preferentially establish connections with individuals quite different from their condition, revealing a feature of criminal networks. 

We use four well-known measures to study our multiplex criminal network, identify their roles, and detect potential candidates in rupture strategies. Fig.~\ref{fig:fig3_mesuresnetwork} shows the results of the four measures for each node.

\begin{figure}[ht!]
\centering
\includegraphics[width=11cm]{fig3_mesures_network.png}
\caption{Measures of degree centrality, betweenness centrality, multiplexity and node diversity.}
\label{fig:fig3_mesuresnetwork}
\end{figure}

Considering the four highest values in decreasing order, we have, for degree centrality, individuals $80$, $48$, $51$, and $47$; for betweenness centrality, individuals $80$, $47$, $48$, and $21$; for multiplexity, individuals $48$, $21$, $47$, and $80$; and finally for node diversity, individuals $48$, $80$, $47$ and $21$. Degree and betweenness centrality are measures that could be related to a position of power in the criminal network. Multiplexity could be related to trust since the individual who participates in several crimes could indicate a closer relation. Node diversity could be related to readjust, and network resilience since a great variety of connectivity configurations shows more options to substitute or change individuals and roles. The node diversity analysis revealed interesting new features as important roles not identified by traditional measures of centrality. One example is individual $48$, identified as the node with the highest diversity value, while the individual $80$ is the one with the highest centrality. However, individual $48$ possesses third place in the centrality ranking. Individual $48$ is interesting since it is the one that participates in the highest number of crimes, contains an increased number of redundant links with node $80$. Another important fact of this node is that it is the only individual identified by the studied measures participating in the arms trafficking network. Then, it is very likely that the presence of individual $48$ guarantees the proper functioning of the multiplex criminal network, having perhaps, a more important role than individual $80$.
%
\subsection{Rupture of Criminal Networks}
%
Criminal networks are flexible and difficult to rupture, mainly because they are dynamic. Changes in the crime network can occur for several reasons, as is the case competition, stagnation, adaptation etc. The effects of network disruption can then be understood within an adaptive dynamic. There are three situations that destabilize a crime network: reduction in the flow rate of information, reduction in the ability of the individuals to conduct their tasks, and failure in the decision-making process.

Network rupture can be understood as a network state in which information, assets, and knowledge cannot be disseminated efficiently \cite{duijn2014}. As a result of an outage, criminal networks need to adapt to this change. This is conceptualized as network resilience. Resilience consists mainly of two abilities: first, to absorb and thus resist rupture; and second, to adapt, when necessary, to the changes resulting from that rupture \cite{duijn2014}. The rupture of crime networks is done by removing strategic individuals in the criminal network. Therefore, it is necessary to understand the possible adaptation mechanisms after these removals. Then, the ability of the organization to regain structure after implementing a rupture strategy needs to be simulated, as explained below.

A higher-ranking individual in the strategy classification is withdrawn and is replaced by the next element in the ranking. After, this new actor assumes the bonds of the substituted individual. This process is repeated until the disruption of the network. Disruption occurs when all nodes have only one edge with the central individual, and this individual is removed from the network. Thus, in modeling the resilience of the criminal network, we show the dynamic interaction of the criminal network in the search for adequate substitutes for network resilience. At each step, an individual is guided by the chosen rupture strategy. The impact of the strategy is then measured. We use here the network efficiency measure proposed in \cite{duijn2014}, given by Equation \ref{eq:efficiency},


\begin{equation}
    E(G)= \frac{1}{n(n-1)}  \sum_{(i<j \in G)}{ \frac{1}{d(i,j)} }, 
    \label{eq:efficiency}
\end{equation}

in which $G$ is the network, n is the number of individuals in the $G$ network, and $d(i,j)$ corresponds to the distance between the $i$ and $j$ individuals of $G$. The denominator creates a measure ranging from 0 to 1, where 1 is complete connectivity, and 0 represents that the components of the network are entirely isolated. This metric shows how efficiently individuals can communicate and interact.

The efficiency $E$ of the network is measured at the time of disruption, when an individual is removed, and when another individual from the network assumes its place, thus demonstrating the disruption efficiency and resilience efficiency of the network.

To demonstrate the dynamics of rupture and resilience in the criminal network, nine strategies for network disruption are presented, each one taking into account one of the following criteria: degree centrality; betweenness centrality; multiplexity; node diversity; multiplexity/degree; multiplexity/betweenness; multiplexity/diversity; random choice; and directed by key roles (Fig.~\ref{fig:fig4_keyrolesstrategy}). Other simulations were carried out, but they did not obtain good results for criminal network disruption.

\begin{figure}[ht!]
\centering
\includegraphics[width=8cm]{fig4_keyrolesstrategy.png}
\caption{The directed strategy of key roles uses the ranking of measures in order of importance to identify key roles in criminal networks: node diversity (a), betweenness (b), and degree (c). The ranking of key roles is then established: an actor is selected by the highest ranking of (a); the next actor is selected by the highest ranking of (b), if not selected before; the next actor is selected by the highest ranking of (c), if not selected before; selects the next actor in (a), if not selected before; selects the next actor in (b), if not selected before; selects the next actor in (c), if not selected before. Continue until the end of the selection of all actors. The image is an example of the selection of actors by the directed strategy of key roles.}
\label{fig:fig4_keyrolesstrategy}
\end{figure}

Fig.~\ref{fig:fig5_resilience} illustrates the efficiency measures obtained in the simulations for each applied strategy. An efficient network implies that the network is highly resilient and flexible, being able to restore its network structure. This allows us to say that the network can recover by finding close and known substitutes in the crime network.

\begin{figure}[ht!]
\centering
\includegraphics[width=11cm]{fig5_resilience.png}
\caption{Efficiency resilience in each strategy until the moment of rupture is applied in the crime network. The strategy of key roles is identified as the best by breaking in fewer steps.}
\label{fig:fig5_resilience}
\end{figure}

Each simulation was carried out to disrupt the criminal network, with the strategy directed to the individuals in order of classification until the moment of network disruption. The disruption of the network occurs when, being all nodes connected only to the central node, this central node is removed. The number of steps needed for the disruption of the network for each strategy is the following: $73$ for degree centrality; $64$ for betweenness centrality; $71$ for multiplexity; $68$ for node diversity; $62$ for degree/multiplexity; $67$ for betweenness/multiplexity; $67$ for diversity/multiplexity; $77$ for random; and $59$ for direct to key roles. Therefore, the best strategy for disrupting the criminal network is the one directed to key roles, as it promotes a disruption with the fewest numbers of individuals removed.

To better understand the process, let us comment on the particular case of individual 51. In the multiplex study, he loses importance as a hub in some measure for having committed only the crime of drug trafficking, being part of only one multiplex layer. Despite this, he is an important individual and organizes a community on the network. Using the degree, betweenness, multiplexity, node diversity, multiplexity/degree, multiplexity/betweenness, multiplexity/node diversity, random and key roles strategy, this individual assumes the central role, respectively, in $2^o$, $27^o$, $68^o$, $66^o$, $53^o$, $52^o$, $64^o$, $58^o$ and $3^o$ resilience step, which explains the behavior exhibited in the resilience efficiency of the network, when this individual is removed (Fig.~\ref{fig:fig5_resilience}).

Even with the best strategy, which is directed to key roles, the human and material cost for security intelligence is very high and impracticable.

A viable approach would be the intervention of public security intelligence to stop crime and criminality, given the fundamental role they play. So, we use the key role strategy to disrupt the arms trafficking layer since this layer guarantees the proper functioning of the multiplex criminal network. The disruption efficiency measure was established using the network efficiency equation \cite{duijn2014} at the moment of disruption. To disrupt the layer of arms trafficking, we use the ranking of the strategy directed to key roles. Removing the first 12 individuals by ranking utilizing this strategy (or the possible number if the layer has less than 12 individuals), the disruption reduces the number of individuals and links of the multiplex layers (Fig.~\ref{fig:fig6_rupture}).

\begin{figure}[ht!]
\centering
\includegraphics[width=11cm]{fig6_rupture.png}
\caption{Disruption efficiency in each layer of the criminal network, using the first individuals to rank the targeted strategy key roles. With this targeted strategy, the layers of ideological falsehood and arms trafficking are disrupted, and others have their efficiency reduced.}
\label{fig:fig6_rupture}
\end{figure}

After applying the targeted strategy at the first key roles, the result in the crime layers was the following (Fig.~\ref{fig:fig7_crime_multiplex_ruptura}):

\begin{figure}[ht!]
\centering
\includegraphics[width=11cm]{fig7_crime_multiplex_ruptura.png}
\caption{After the rupture, the multiplex criminal network represented in the structure are: ideological falsehood (0 edges/0 nodes), arms Trafficking (0 edges/0 nodes), kidnapping (3 edges/3 nodes), weapon Carrying (7 edges/10 nodes), homicide (11 edges/12 nodes), theft (14 edges/17 nodes) and drug trafficking (19 edges/25 nodes); a total of 53 links and 37 individuals.}
\label{fig:fig7_crime_multiplex_ruptura}
\end{figure}

\begin{itemize}
    \item The layers of ideological falsehood and arms trafficking were completely interrupted.
    \item Kidnapping’s crime layer was left with a group of three individuals.
    \item The weapon carrying crime layer was split into four groups.
    \item Homicide’s crime layer was split into three groups.
    \item Theft’s crime layer was split into six groups.
    \item The drug trafficking crime layer was split into nine groups.
\end{itemize}

The efficiency of the rupture strategy becomes more evident by comparing Fig.~\ref{fig:fig7_crime_multiplex_ruptura} with Fig.~\ref{fig:fig2_crimemultiplex}.

%
\section{Conclusion}
%
We seek to simulate and understand the disruption and recovery of a criminal network in multiplex layers through measurable numerical data. Nine disruption strategies, based on degree centrality, betweenness centrality, multiplexity, node diversity, and random choice, were applied, and the results were presented. The strategy based on key roles was also used, showing excellent results, breaking through the targeted layer, and reducing and dividing the other layers of crime. Although the criminal network is flexible and adaptable, a disruption directed to key roles can be effective and allow police intelligence operations to be targeted to remove key criminals, with a conscious effort to intervene in the long term persistently.

With the proposed method of disruption based on key roles, we believe that the police have gained an important tool to identify, visualize and qualify criminal activity in near real-time, increasing the chances of a successful criminal network disruption.

We hope to apply this method to other police databases and, eventually, integrate it with social data to assess possible hidden connections between individuals and hidden key actors to create a model that allows better identification of key roles and network interruption. The use of machine learning models in this scenario can also add to the identification of hidden connections and individuals, expanding the focus of the metrics used.

\backmatter


\section*{Declarations}

\subsection*{Disclosure of potential conflicts of interest}

The authors declare no competing interests.

\subsection*{Research involving human participants and/or animals}

The research did not involve humans or animals.

\subsection*{ Informed consent}

The research respects the highest ethical and moral standards established by the academic community. All data were provided for research by the 'Military Police of Minas Gerais'/Brazil, and were treated and anonymized in accordance with brazilian law.



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