Аннотация и ключевые слова
Аннотация (русский):
The report is devoted to research on the capacity of information networks and related stochastic fluctuations and bursts. In open systems, the exchange of energy and information with surrounding bodies, due to their complexity, generates the formation of various structures. This process of creating structures is especially relevant when it comes to systems with a fractal structure. Analysis of such processes should be carried out in terms of fractional geometry. The dynamics of such processes are characterized by such effects as memory, complex spatial mixing processes and self-organization. The use of fractional dynamics methods opens up new possibilities for solving problems of forecasting and decision making in complex systems

Ключевые слова:
Stochastic processes, entropy of Tsallis, branching processes, percolation, Deep Learning
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It is known that during the operation of complex transport systems (networks) both the capacity of the network (channels) and the demand for traffic are subject to stochastic fluctuations and bursts (Levy flight) [1-5]. These random fluctuations are the main sources of uncertainty of the transit time, and a result, losses information (technological losses).

It is important to note that the heterogeneity of stochastic processes and the asymmetry of claims causes uncertainty, different from the traditional one. In the context of the coordinate of emerging problem, for the purpose of traffic assignment, the model of entropy of Tsallis is proposed that allows tracking coherent processes.

It is known that stochastic transport processes represent a generalization of the diffusion process, which is expressed in the transition from the usual root dependence to the ratio [1]:

,                                                                           (1)

characterized by dynamic exponents  (here  - the coordinate of the wandering particle,  - time).

When sub diffusion, the presence of traps leads to a divergence of the average waiting time for jumps , so that the latter acquire a discrete character in space and the transport process slows down ().

Its acceleration in the process of super diffusion of levels is due to the fact that the particle at discrete instants of time performs jumps of arbitrary length, characterized divergent mean square displacement  [1].


  1. Mathematical Model of Fractional Traffic Levy Motion

 - stable Levy motion, . The in terms of the Riemann-Liouville operator, we have [2, 3]:

,                                 (2)

where  is the ordinary symmetric  - stable Levy Motion (oLm), and  denotes the gamma function,  - Hurst parameter.

From a mathematical model of fractional traffic Levy will be expressed as [2, 3]:

,                                         (3)

where  is the mean input rate,  is the scale factor, and  is the fLm process defined by (2).

The model has four parameters  and  with the following interpretations [2, 3]:

  •  is the mean constant input rate
  •  measures the “thickness” of the tails of the stable distribution
  •  is the scaling parameter that can be seen as the dispersion around the mean of the traffic
  •  is the Hurst parameter (index of self-similarity)

Based on the mathematical model of traffic (3) transport models of the types: Branching processes as a particle branching and fractional Brownian motion (fBm).



Figure 1. Levy motion.


    1.  Branching processes as a particle branching

In this section deals with general Bienayme-Galton-Watson processes describing branching particle systems in the discrete time setting [6, 7]. We denote by  the number of -th generation particles whose types belong to . The same generation particles are assumed to produce offspring to a random algorithm.

A key characteristic of the multi-type reproduction law is the expectation kernel [4]:

 ,                             (4)

where the operator  is indexed by type  of the ancestral particle.

Here -processes, branching particle systems are characterized by general linear-fractional () distributions.

It is assumed that the type of the desired genus , the total number of offspring  follows linear-fractional distribution [6]:


where .

If , where  - is kernel, the ancestral particle has no offspring, and with probability , it produces a shifted-geometric number of offspring [6]:

,                                          (6)

where parameter  is independent of .

Then on the basis of [7] the model of fractional traffic Bienayme-Galton Watson processes will have the form:

,                                                  (7)

where - scale factor.


Figure 2. Branching process.                                                Figure 3. Percolation lattice.


    1.  Fractional Brownian Motion (fBM)

Fractional Brownian motion is defined by its stochastic representation [8]:

,        (8)

where  represents the gamma function   and  is called the Hurst parameter. The integrator  is a stochastic process, ordinary Brownian motion. The traffic model using Brownian motion is defined: .

Figure 4. Fractional Brownian motion.

  1. Transport problem on a percolation lattice (algebraic structures)

In report shows the possibility of homomorphism stochastic processes in percolation lattice in the context of recognizing the transport properties of these systems. The formal basis for embedding systems is the results of a modern general algebra on the embedding of complex algebraic structures into relatively simple algebraic structures.

In this connection, the principle of fractal homomorphism (universal similarity), in the context of category theory, fixes on the one hand the fundamentality of Not What is reflected, but How, and on the other hand means the mutuality of fractional structures of any scale [7, 9].



Figure 5. An example of the solution of the transport problem.


    1.  Main provisions

Percolation represents the basic model for a structurally disordered system. The percolation transition is characterized by the geometrical properties of the clusters near . The probability  that a site belong to the infinite cluster is zero below  as [9, 10]:

.                                                    (10)

When  approaches ,  increases as [8, 9]:

,                                                     (11)

with the same exponent  below and above the threshold and  - correlation length.

Here  depends on the type of the lattice, the critical exponent and , and  they are universal and can be depend only from the dimensions of the lattice.

Axiom of embedding. Let the one – dimensional array 1D  be transformed into a square matrix 2d, .

The fractal dimension  of the analyzed segment of the array is empty.

Then the homomorphism  will be determined as:

,                                     (12)

 for reliability.

Then the percolation lattice will represent the geometric and dynamic realization of the stochastic cluster.


    1.  Conductivity of Percolation Lattice

It is noted [11, 12] that the conductivity is represented as:

,                                                       (13)

where the critical exponent  is (semi) – universal,  - critical probability.

For percolation on a lattice,  depends only on , where  is lattice dimension.

Critical exponent for two lattice dimension equals .

Thus, a transport problem is posed in the context of the homomorphism of stochastic discrete systems onto the percolation lattice.



As a result of the analytical and numerical studies, it can be concluded that it is necessary to take into account a large number of accompanying and influencing parameters of the information network. This approach will allow you to more reliably assess the resources of the existing network and help in choosing the best configuration. Comprehension and application of a large amount of important visual information requires the use of Visual Thinking technology, as well as the use of a set of Deep learning algorithms.

Research funding

The work was carried out by the author as part of his scientific research and was not funded by grants or third-party sponsorship.

Список литературы

1. Bender C., Lindner A., Schicks M. Finite Variation of Fractional Levy Processes. Journal of Theoretical Probability/ 2012, Volume 25, Issue 2, pp. 594-612.

2. Laskin A., Lambadaris I., Harmantris F., Devetsikiotis M. Fractional Levy Motion and Its Applications to Network Traffic Modeling. Department of Systems and Computer Engineering, Carleton University, Ottawa, Canada, 2001. pp. 1-18.

3. Vladimirsky E. and Ismailov B. Fractional Structure «MIXING - TRANSPORT» as an Open System. Eastern-European Journal Technologies. Vol. 4, № 4 (70). 2014. pp. 4-9.

4. Federbush A. and Kantor Y. Percolation perspective on sites not visited by a random walk in two dimensions. Physical Review E, Vol. 103, Issue 3.032137. 2021.

5. Burlak G. and Medina-Ángel G. Applications of a neural network to detect the percolating transitions in a system with variable radius of defects. Chaos. 2020 Aug;30(8):083145. doi: PMID: 32872808.

6. Lindo A. and Sagitov S. General linear-fractional branching processes with discrete time. arXiv: 1510.06859v2 [math. PR]. 2016. pp. 1-17.

7. Sagitov S. Linear-fractional branching processes with countably many types. arXiv:1111.4689v3 [math.PR] 13 Dec 2012. 24p.

8. Mandel Brot and van Ness J.W. Brownian motion, fractional noises and applications. SIAM Review, 10, 1968. pp. 422-437.

9. Vladimirskij E.I. Fraktalnost algebraicheskix struktur. trudy iv mezhdunarodnoj konferencii «identifikaciya sistem i zadachi upravleniya» SICPRO. Moskva. 2005. s. 492-502.

10. Feder J. Fraktals. M.: Mir. 1988. 254p.

11. Bunde A. and Kantelhardt W. Diffusion and Conduction in Percolation Systems: Theory and Applications. / 4805-springer-Diffusion-pdf. 2005. pp. 1-19.

12. Olemskyj A.I. Ierarxicheskaya kartina superdiffuzii. Pisma v ZHTFb, Tom 71, Vyp. 7, 2000. s. 412-417.

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