About Modeling Digital Twins of a Social Group

Kovalev Sergei ORCID: 0000-0002-1132-6888 PhD in Technical Science Associate Professor, Department of Computer Engineering, I.N. Ulianov Chuvash State University 428015, Russia, Republic of Chuvashia, Cheboksary, Moskovsky Prospekt, 15, office B-309 srgkov@gmail.com Smirnova Tatiana ORCID: 0000-0001-6687-9415 PhD in Physics and Mathematics Associate Professor, Department of Mathematical and Hardware support of information Systems, I.N. Ulianov Chuvash State University 428015, Russia, Republic of Chuvashia, Cheboksary, Moskovsky Prospekt, 15, office B-304 smirnova-tanechka@yandex.ru Filippov Vladimir ORCID: 0000-0002-7240-4405 PhD in Physics and Mathematics Associate Professor, Department of Mathematical and Hardware support of information Systems, I.N. Ulianov Chuvash State University 428015, Russia, Republic of Chuvashia, Cheboksary, Moskovsky Prospekt, 15, office B-304 filippov_v_p@mail.ru

Andreeva Antonina ORCID: 0000-0003-0843-2230 PhD in Technical Science Associate Professor, Department of Computer Engineering, I.N. Ulianov Chuvash State University 428015, Russia, Republic of Chuvashia, Cheboksary, Moskovsky Prospekt, 15, office B-309 antonina-andreeva21@yandex.ru

IntroductionA digital twin is a software analogue of a real object ^[1] that simulates internal processes ^[2], technical characteristics and behavior under the influence of interference and the environment ^[3].

This concept is part of the fourth industrial revolution ^[4] and is designed to help detect problems faster, find out what will happen to the original in different conditions and, as a result, produce better products ^[5]. The topic of the digital double is becoming popular in various subject areas, for example, it is considered in intelligent battery management systems ^[6], finds application in information modeling of parts based on ontology ^[7].

Currently, research is being actively conducted on mathematical modeling of the behavior of social groups (students, etc.) and their decision-making depending on emotional upbringing and logical experience.

Mathematical models are created according to scientific theories, including the general theory of human psychology.

This allows you to create digital doubles similar to real social groups of people, rather than fictional abstract creatures.

Numerical values are used as input "psychological" parameters of models that allow "calculating" the behavior of social groups. To describe the psychological behavior of social groups of people, the input parameters of mathematical models of a digital double are numerical characteristics inherent in a person (student).

Population algorithmsIn this article we will focus on the applied aspects and present some results of the mathematical theory of digital doubles of social groups (students).

To solve the problem of creating a digital double of a social group (students) as one of the tools, it is advisable to use technologies of population algorithms (evolutionary algorithms; algorithms using the concept of swarm algorithms; algorithms based on other mechanisms of living and inanimate nature) ^[8].

Swarm algorithms are a class of algorithms that appeared on the basis of observations of colonies of living creatures: flocks of birds, schools of fish, swarms of bees, colonies of ants ^[9].

The methodology of swarm algorithms is based on decentralized systems consisting of monotonous elements (agents) indirectly interacting with each other ^[10] and with the environment to achieve a predetermined goal ^[11]. It is this definition that underlies the swarm algorithm ^[12].

Swarm algorithms (part swarm algorithm, ant algorithm, etc.) arose as a result of modeling the behavior of birds in a flock ^[13] (part swarm algorithm ^[14]) and studying the principles of ant behavior in nature (ant colony algorithm or ant algorithm ^[15]). In ^[16], the proposed modification of the particle swarm algorithm is also justified using the theoretical concept of swarm intelligence, and not based on mathematical models of the algorithm. After the spread of swarm algorithms, various mathematical models were applied to them, including Markov chains. The use of Markov chains makes it possible to prove the convergence of the algorithms under consideration to the global optimum only theoretically, with the running time of the algorithm tending to infinity, but this does not explain the high efficiency of swarm algorithms shown in numerous experiments and for solving practical problems with time constraints.

The changes taking place in a group of students is an example of the collective behavior of a social group. They can move in a coordinated manner, split up and then reassemble into a group.

It was noticed that groups of people (students) often solve optimization problems, usually multi-criteria.

In the course of the work, it was decided to focus on the particle swarm method (the swarm part algorithm) and check how effective and convenient this method is to implement, and how it can be improved if such a need arises.

The classic rules of object behavior were formed by Craig Reynolds back in 1986 from his bird watching. With their help, he created a computer model of the flock, called boids. Here are the basic rules.

1. Each bird tries not to approach its relatives by less than some minimum allowable distance. This rule is designed to avoid collisions among birds.

2. Each bird tries to choose its own velocity vector so that it is closest to the average velocity vector of all birds in its local neighborhood. This rule coordinates the direction and speed of the birds.

3. Each bird tends to be located in the geometric center of mass of its local neighborhood. This rule forces each bird to stay inside its flock.

Application of swarm algorithms technology in modeling digital twins of a social group

Fig. 1. Graphical simulation of a group in Java Script using the library three.js

Figure 1 shows a graphical simulation of the group in the Java Script language using the three library.js, which is used to create and display animated 3D computer graphics in the development of web applications ^[17].

As the social group under study (a group of students) finds the optimal position in space, so the element of the digital twin of the particle swarm model based on them can search in space, in particular, the extremes of functions. Which, for example, is applicable to finding the minimum of the loss function in machine learning.

Students in the group move in two-dimensional space. Each person has a velocity vector, an acceleration vector, and a position vector. We will also introduce the concept of a local neighborhood of a person (the object of research), within which the student controls his position relative to the rest of the members of the social group.

The key value in the particle swarm method is an algorithm for updating the speed of movement of an individual. For example, by the formula

v[i] = v[i] + a(p[i] - x[i]) + b(g - x[i]),

where:

• v[i] – human speed;

• p[i] is the simplest individual memory equal to the coordinates of the best point of the particle trajectory for the entire time of its existence;

• g is the collective memory representing the coordinates of the best point reached by the whole swarm;

• x[i] – coordinates of the particle;

• a and b are some coefficients, usually from the range ^{[0; 1]}; if parameter a is chosen randomly, then b = 1 – a.

Here is the search algorithm.

1. A group of M people is created.

2. The vector p[i] and the vector g are calculated.

3. Students' speeds are randomly initialized.

4. The program runs until the stop criterion is met (the target is found with an accuracy that satisfies us).

5. For all students: choose random coefficients a and b in the range from 0 to 1.

6.  v[i] = v[i] + a(p[i] - x[i]) + b(g - x[i]).

7.  x[i] = x[i] + t*v[i].

8. If f(x[i]) > f(p[i]), then p[i] = x[i].

9. If f(x[i]) > f(g), then g = x[i].

10. We return the vector g (the assumed coordinates of the target).

In such an algorithm, the speeds increase uncontrollably, so you can enter the resistance of the medium for simulation.

v[i] = G*v[i] + a(p[i] - x[i]) + b(g - x[i]),

where R < 1 is the resistance coefficient of the medium. Or you can simply set the maximum speed of a person, above which she will not be able to accelerate.

The acceleration vector is formed from the results of three main functions:

• alignment – alignment;

• cohesion – cohesion;

• separation – separation.

These functions correspond to the three classical rules of behavior of the boids model. The results of the functions (vectors) add up and form the acceleration vector of a person.

During the development of the digital twin model element of a social group (students), the Entity Component System design pattern described by the Entity-Component-System structure was applied. Entities are containers for components. Components store all kinds of properties of events or objects (students). Systems determine the way they are processed and store methods for their execution.

 Fig. 2. Entity Component System Design Pattern

Data processing was performed using the C# Job System, which provides parallelization of computing processes and is integrated into the Entity Component System. Computational tasks are formed for each student, which are distributed across computing processors, which gives a significant increase in productivity.

Fig. 3. Parallelization of the social group model (students) using JobSystem

In the course of the work, a program simulating a social group (students) was implemented as one of the constituent elements of the digital twin of a social group.

Swarm algorithms do not require the creation of new populations at each step by selecting and crossing agents of the previous population, but use collective decentralized movements of agents of one population, without selection procedures, destruction of old ones and generation of new ones ^[18].

Swarm algorithms are promising in the field of practical application. On their basis, it is possible not only to solve the problems of digital twins, but also to control groups of robots ^[19], robotic systems and complexes ^[20].