Maths vs Cartels. How basic algebra can be used to combat Mexican organized crime

Redlog for Borderland Beat

The following article tries to explain how the use of mathematics through graph theory and Network Analysis can be useful for defeating Mexican criminal structures. The purpose of the text is purely cultural, so it´s not designed as a mathematical essay but a simple piece that might help everyone to better understand complex issues. Hope you find it useful
One common mistake when taking about Mexican criminal, the panorama is to understand the criminal structures acting all over the country as monolithic entities labeled generically as “cartels´´ The term cartel was applied for the first time to the criminal panorama by Colombian writers and journalists trying to explain how different illicit structures engaged in different activities had joined forces to harvest, produce, export, and sell cocaine internationally. 
Since the 1980s the term cartel has been used by thousands of authors, academics and journalists to describe criminal associations acting both in North America, Central America, and South America. The great error when using such a term is to think that a certain cartel is a unique structure composed of a tight network comprised of very correlated individuals who act uniformly receiving orders and structures from an upper echelon made of a little group of heavy hitters, the real leaders of the groups. 
Thus, when talking about the Venezuelan Cartel de los Soles we might think that Nicolas Maduro, Diosdado Cabello, and Tareq El Aissami are the ones directing the group and giving orders to the Fuerza Armada Bolivariana to buy cocaine from Colombian and Brazilian wholesalers and then use their connections with Central American smuggling groups to send the product by air to Mexico. 

Applying the same logic we would tend to think that the Sinaloa cartel is managed on the top by Ismael Zambada and the sons of Joaquin Guzman Loera which would be under the control of the nearly dozen of groups linked to the groups acting on the north, east, and center of Mexico. This interpretation of criminal groups in general (and Mexican organizations in particular) is a simplistic generalization that has created a mantra (the term “cartel´´ by itself) which has done a disservice for several Latin American countries since the designation of every single criminal group acting in Mexico or Colombia as “cartels´´ has facilitated the current war on drugs which until now is a total disaster.
Actually, the real structure of criminal groups is very far from being unique and monolithic, like formal armies, and resembles a form of organization that we could define with one word: network. A network can be defined as a group of subjects or actors which are related to each other with a certain common purpose. This purpose is extremely wide and can represent anything: friendship, love, kinship, money, electricity, etc. Networks are studied by several branches of mathematics which include algebra, statistics, probability, or geometry, and have created two major theories: graph theory and network analysis. Although network analysis is the most important one and is used by intelligence agencies for tackling criminal and terrorist groups in this article we´ll be using mostly concepts and techniques coming from graph theory since they are simpler and easier to handle.
Graph theory was formulated by the first time in the XVIIIth century the Swiss mathematician Leonard Euler when he tried to solve a problem which consisted of crossing seven bridges of the city of Konigsberg connecting three pieces of land without crossing the same bridge twice. Since then graph theory has experienced an incredible array of innovations and improvements which have made Graph Theory a central part of several disciplines such as computation, architecture, or social/urban planning. Graph theory is centered on the analysis of the basic characteristics of networks. To apply it to the Mexican organized crime panorama we must know about the basics characteristics of networks.
A network is composed of three different elements:
– Nodes: they represent different subjects which are interconnected with the purpose of achieving a
        common goal
– Links: they represent the relation between two nodes and are depicted with straight lines

– Flow: it tells the direction of a certain link. It can be unidirectional or bidirectional. 

Imagine the following example. In the State of Puebla, there´s a local criminal group operating an oil theft network. The mechanics of their operations are the following. The criminal group pays a Pemex officer for information about the routes of tank trucks. The driver of the trucks has an agreement with the Pemex officer, he´ll receive money from the officer and all he has to do is to give the criminal group half of the cargo. 
The group will sell the gas stolen from the tank truck to a local businessman owning a gas station, who will pay the group for the gas. In the end, this operation (which is repeated each time the criminal group steals a cargo of gas) represents a network composed of four nodes (criminal group, Pemex officer, truck driver and gas station owner) which are interconnected by 5 different links through which two items (gas and money) flow unidirectionally (between Pemex officer and truck driver) or bidirectionally (between the criminal group and the gas station owner in this case)

This is the basic explanation of how a quite simple huachicol network would work. In fact, there are very few networks like these since most Mexican criminal groups have been absorbed by bigger organizations. Nowadays most Mexican criminal groups are very sophisticated and involved in a plural business portfolio. A more rational representation of a local criminal group would be the following.
Now we must analyze the basic characteristics and properties of networks, which are comprised of several network indicators. Take into account that I´ve simplified the information very much, in fact, things are more complicated.
The first thing we have to analyze when confronting a network is its density. The density of a network can be defined as the degree of homogeneity or cohesion it portraits. It´s calculated with a mathematical formula expressed as:
Although this might seem complicated it is not. The formula can be explained as the result of the existing combination between nodes divided by the possible total combinations being multiplied by 100. The higher the density of a network, the more developed a network is and the harder it will be to dismantle it.
Another important characteristic of networks are four concepts we know as centrality measures:
– Centrality degree: it´s an estimation of the number of neighbors that a node has and reveals which nodes are the most important. Thus, a node with a high degree of centrality will be connected to a lot of nodes while one with a lower degree will be tied few. Applying this logic to the organized crime panorama is obvious that attacking nodes with high centrality degree (jefes de plaza or high-ranking lieutenants) will be more useful than attacking nodes with lower degrees that are at the bottom of the network (sicarios, balconies or associated personnel)
– Centralization index: it is used for identifying the network´s central node. In most networks, we can find a node that is the central subject of the whole system. In criminal groups, centralization index might help to track the individual where most links gather thus revealing fundamental actors which might or might not be the real bosses behind the group
– Betweenness: it reveals how important a node is inside the network indicating the frequency with which a node acts as a “bridge´´ between two nodes. In other words, it identifies the nodes that can control the flow of information or resources through the network.
– Closeness: it indicates how accessible a node is inside the network by indicating the capacity of a node to reach the rest of the nodes of the network.
The graph theory applied to the Mexican panorama:
If we want to fight Mexican criminal groups effectively, the use of graph theory and social network analysis is fundamental. Since 2006 the current low-intensity conflict that has erupted in Mexico has been fought erratically, but we can identify a clear trend or strategy applied by the Government: the identification of key leaders of criminal organizations, their inclusion in top wanted list and their frantic search, which might end with the boss imprisoned or killed (with a few mythic exceptions such as El Mayo, Caro Quintero or El Mencho) The direct consequence of this combat strategy, which has received the name of “Kingpin Strategy´´, has been the decapitation of the big criminal groups operating during the early 2000s.
Due to the capture or killing of the high-ranking bosses the groups (which were always alliances between dozens of different entities or factions organized under a mutual name/brand) fragmented and imploded into dozens of more little groups which almost immediately started fighting for the control of every available market niche (legal or illegal) The absence of a network analysis tool in the Mexican strategy in order to identify the fundamental and vital individuals inside the criminal groups (which aren´t the high bosses necessarily) has led to the increasing violence and the expansion of the criminal structure all over the country.
Nevertheless, although the Mexican armed forces failed to apply the network analysis to their war against organized crime there are other countries where Social Network Analysis (the successor of graph theory) is being used for combating terrorist and criminal threats. The US is a great example.
The US Department of Defense has created and commercialized several software tools specially designed for the gathering of data, its analysis, and its graphic projection through the use of drawing programs. Thus, software programs such as Palantir Government, UCINET or ORA have been (are being) used for representing graphically the clandestine structures of criminal groups. This representation makes it easier to identify the vital nodes through which most of the information or resources of the organization are channeled. Its surgical elimination facilitates a strategic debilitation of the whole network and ultimately its demise instead of causing a simple division of the initial network an indefinite number of new ones due to the erratic elimination of cohesive nodes (the bosses) that provide the network with the necessary homogeneity but who are not vital for its existence.
For example, a brilliant essay by two students at the Monterrey Naval Postgraduate School used social network analysis for drawing the clandestine structures of Los Zetas by 2011. They used Palantir and UCINET for analyzing the data and ORA for drawing the networks.
This is an image of the structure of Los Zetas right before the death of Heriberto Lazcano in 2012. As we can see what has been mentioned thousands of times as Los Zetas cartel is in fact a very complex structure composed by where the bosses are located several clusters composed by dozens of people linked to the hardcore by several vital nodes. The clusters aren´t anything but affiliated groups while the vital nodes are the commanding individuals of these groups. Now, a consideration about what happened with Los Zetas’ leadership might bring some light to the current state of affairs.
Between 2010 and 2015 Los Zetas became the main target of the Mexican armed forces. Their ruthless tactics, widespread violence, and expansionist strategies made them public enemy number one.
By 2011 their main leaders (the original Zetas, not more than 40/50 individuals) were mostly identified and seriously prosecuted. In less than 3 years most of them had been either captured or killed. This caused the dispersion of several cells/groups which had been tied to the organization by these people. The Cartel del Noreste, Los Talibanes (although they were the first offshoot of Los Zetas), Zetas Sangre Nueva, Zetas Vieja Escuela, etc. All these groups became independent because the links that tied them to the whole structure were severed when the original Zetas disappeared.
This image represents (in red) the 10 Los Zetas members with the highest centrality degree of the network. Thus, according to the software these people were the most important Zeta leaders by 2011 although there are some singular omissions such as the Treviño Morales brothers and others which might explain some inconsistency due to the absence of information when the study was carried but more or less we can identify these actors as vital nodes of the network. The centrality degree of these 10 top Zetas are the following:
In conclusion, these 10 individuals were the core of Los Zetas according to the software used. From all of them, only three (in yellow) are still at large. The rest are either dead or imprisoned. What was the direct effect of the elimination of these people? As we have said, the disaggregation of the initial tight network into several smaller groups, the clusters seen in the image. These people were like the knots holding a tied group of elements, once the knots were dissolved the elements once tied became disaggregated and started operating independently.
What could have been done in order to prevent this massive criminal explosion?
In my opinion, instead of targeting the maximum leaders the Mexican should have eliminated not the high-ranking but the medium level nodes. This is the individuals in command of the clusters who were in contact with the high-level nodes. The importance of these cluster commanding nodes is revealed by the Betweenness which as we have said indicates how important a node is depending on the frequency with which it acts as a “bridge´´ between two other nodes.

Highest graph measure of centrality (Betweenness) among Los Zetas nodal structure
As this image shows the cluster commanding nodes (which are the commanders of groups affiliated to the whole Los Zetas´ structure) have the highest Betweenness degree of the whole network since they are fundamental for connecting the bottom structures with the high command. Their elimination should have been a priority as important as the elimination of the higher bosses since their role was fundamental for the daily functioning of the whole structure. Without them, Los Zetas could have subsisted, but with a much tinier network since the symbiosis between local groups and higher bosses would have been much more difficult to be achieved.
What would have been achieved with the elimination of these nodes of high Betweenness? Of course, a basic disaggregation of the different groups comprising the Los Zetas structure, but with a very different effect. In my opinion a great number of local criminal cells devoted to different activities such as extortion, oil theft or murder would have found the daily conduction of their activities much more difficult and since most of them weren´t carrying violent activities but were merely simple members of a criminal group the level of violence would have been significantly lower and the dismantling of the tinier subsisting clusters would have been much easier without clear and competent leadership.
The software tools mentioned can locate the network members whose elimination would cause the greater disruption of the network.
The following image identifies the 10 most relevant actors categorized as Key figures depending on the degree of damage in terms of disaggregation that their elimination would cause. They are classified within a range of 1 to 10 were 1 is the lowest value while 10 means that their elimination causes the biggest disruption possible.
What could have been done in order to prevent this massive criminal explosion? In my opinion, instead of targeting the maximum leaders, the Mexican should have eliminated not the high-ranking but the medium level nodes. These are the individuals in command of the clusters who were in contact with the high-level nodes. The importance of these cluster commanding nodes is revealed by the Betweenness which as we have said indicates how important a node is depending on the frequency with which it acts as a “bridge´´ between two other nodes.
Thus, surgical elimination of these most disrupting and vital nodes would have caused the following effect:
As can be seen in the image, what initially was a giant network of connected nodes has become a constellation of few and very little groups which cannot subsist a well-launched Government anti criminal strategy.
We must take into account that we are just suggesting possibilities, not analyzing certainties. The disruption of internal clusters through the elimination of medium level or vital nodes (which are not necessarily the bosses of the network) doesn´t cause the demise of the whole structure “per se´´ The defeat of the whole network is going to depend on several variables. Maybe the most important one is the resilience of the network. Resilience can be defined as the ability of a network to receive a blow which enables it to absorb the aggression, reorganize itself, and continue with its previous performance.
The resilience of a network depends on its degree of development. A highly developed criminal group (with multiple actors and highly capable leadership) will be able to absorb the capture of a local or regional boss without collapsing into chaos. Thus, a higher degree of resilience indicates that aggressions or attacks against the network must be carefully studied in order to hit the most delicate nodes. In other words, the more resilient a criminal group is, the more difficult is to destroy it.
In conclusion, graph theory and network analysis could be highly useful for an efficient and developed anti criminal strategy. The sophistication and high resources of criminal groups as well as the increasingly diversified criminal Mexican panorama cannot put up with an erratic and disconnected strategy based only on eliminating the most visible heads of criminal groups because what we know as cartels behave like XXIth century hydras: cut one head and two more will replace it.
– Luis Daniel Vazque Valencia. “Captura del Estado, macrocriminalidad y Derechos Humanos´´
– Enrique J. Reina and Dennis J. Castellanos. “Exploiting Weaknesses: an Approach to Counter Cartel Strategy´´
– Norman Velazquez Alvarez and Norman Aguilar Gallegos. “Manual Introductorio al Analisis de Redes Sociales´´