American scientists have built a simple model that relies on the theory of gel formation and describes the growth of extremist groups, and then tested its work on real examples from social networks. It turned out that extremist groups on the average unite people with similar interests, and to get rid of network extremism by destroying several “instigators” will not work. The article is published in *Physical Review Letters* , it is briefly reported by *Physics* , the preprint of the paper is posted on the website arXiv.org.

Most terrorist attacks are organized by well-organized groups of dozens of volunteers – for example, the People’s Will, the Irish Republican Army, Al Qaeda and many other organizations. The ways to combat such organizations are relatively well known: it is possible to track suspicious telephone conversations or messages on the network and predict the expected date of the terrorist attack (the members of the group are forced to communicate), locate the main forces of the group, or plant or destroy the majority of its participants. On the other hand, in recent years more and more terrorist attacks have been organized by single terrorists ( lone wolves), which are not directly associated with any terrorist group, but are exposed to them and infiltrated by extremist ideas – for example, at the end of 2014, many isolated groups emerged worldwide who organized attacks in Brussels, Manchester, Paris and London. Predicting such acts of terrorism is much more difficult than “ordinary” attacks, and they have become a complete surprise for the special services.

Typically, security agencies are focused on identifying the identity of a suspected single terrorist before he acts, and paying much less attention to growing extremist groups. Generally speaking, people are predisposed to the formation of such groups – it is enough to recall how widespread racism and religious intolerance . Of course, not all members of such a group will organize terrorist acts or somehow manifest themselves in the “real world”. Nevertheless, the timely detection and destruction of groups will greatly prevent extremists from spreading their ideas.

A group of scientists led by Neil Johnson developed a simple model that describes the formation of extremist groups and explains their sharp growth observed at the end of 2014. This model looks like this. Initially, each member of the group consisting of *N* people is assigned a random value of the value *x*_{i} , which is selected from the segment from 0 to 1 and describes the “area of interest” of the person. For example, a tendency to religious intolerance. Then the members of the group begin to interact with each other: at each step two randomly selected elements can form a relationship with each other with the probability *S *_{ij} = 1 – | *x *_{i} – *x *_{j}|. If the interests of people coincide ( *x *_{i} = *x *_{j} ), the connection is formed for sure ( *S *_{ij} = 1); if they are completely opposite ( *x *_{i} = 0, *x *_{j} = 1), the connection can not arise under any conditions ( *S *_{ij} = 0). Thus, the interaction in the group is determined by its homophilia (network homophily), that is, the mutual similarity of the elements. Simply put, the constructed model expresses an intuitive assumption that people with similar interests tend to gather in groups (“the fisherman of the fisherman sees from afar”). In fact, the model describes gelation (gelation) – for example, souring milk.

Using these rules, scientists have constructed a system of differential equations that describe the growth of clusters within a group. Approximation of the mean field makes it possible to simplify the system considerably – in this case each cluster consisting of several elements is considered as a whole, and the interaction of two clusters is described by only one equation that depends on the sizes *M* , *K* and the parameter *F* taking into account their “common similarity” but not a system of *M* × *K* equations. For the uniform probability distribution *x *_{i} along [0, 1], the parameter *F* = ⅔; for the case when all *x *_{i}are equal to each other, *F* = 1 (all people are equal). Solving simplified system, the researchers found that the phase transition – the union of all elements in one group – occurs at a critical time *t *_{c} = *N* / 2 *F* . At the same time, the number of clusters of size *s* depends on time as *n *_{s} ~ *N* × exp (- ( *s* / 2) ( *t* – *t *_{c} ) ^{2} ) × *s *^{-2.5} . The scientists obtained the results analytically confirmed with the help of numerical simulation of a network with the size *N*= 500.

Then the researchers applied the obtained results in practice, considering extremist groups in the social network “Vkontakte”. It is known that the main surge of attacks at the end of 2014 is associated with macroscopic groups (about a thousand people) that began to actively appear at this moment. As of January 2015, 59 such groups were found on the network, containing about 22,000 subscribers and providing more than 46,000 reposts. Despite the fact that the moderators actively resisted this growth, and by mid-2015 almost all the groups disappeared, in the first few weeks of the group grew almost freely, and therefore had to obey the model developed by scientists. In fact, in the data it was possible to distinguish a clearly defined critical point (December 30, 2014), which corresponded to the value of the parameter *F* ≈ ⅓; nevertheless, on average, the value of *F* was close to *F* = ⅔, that is, the groups did indeed unite people with similar interests. The distribution of the number of groups by their size was subject to a power-law dependence with an exponent *s* ≈ 2.46, which practically coincides with the theoretically predicted value *s* = 2.5. The coefficient of determination ( *R*-quadrate) exceeded 80 percent, that is, the probability of correlation between points was more than 90 percent.

Thus, the authors of the article offer a fundamentally new way of searching for extremist groups, which is based on an analysis of their growth rate. In particular, their work shows that online extremism can not be won by looking for and eliminating a small number of “instigators”, since extremist groups are much more complicated. In addition, the work of scientists is useful for studying other social or living systems that combine similar objects.Of course, this is not the first time that physical theories find application in completely unexpected areas. For example, in July 2015, the American mathematician Filippo Radicci applied the percolation theory to assess the overall stability and vulnerabilities of transport networks without using complex and costly computer simulations.