## 4 Network Model
### User Distribution Model  
For a user distribution model, when placing network elements randomly and selecting a fragment of the Hilbert curve as a model, the structure similar to that can be obtained which shown in Figure 5.  
Figure 5 shows the model obtained by randomly placing the network elements (10 thousand nodes) in a square flat area with a square side of 150 m. For better clarification, Figure 5a shows the Hilbert curve, whereas Figure 5b shows only the network elements (nodes). The network density is about 0.44 devices per m\({}^{2}\) if the total service area is considered while the density is about one node per m\({}^{2}\) if the area occupied by the obstacle model is considered. The elements of the fractal theory are still widely used in modeling communication networks, namely when describing a random processes-traffic. For example, in [29; 30; 31], and many other works, the concepts of fractal theory are used to describe the properties of traffic. This application of fractal theory and the description of random series were laid in the works of [PERSON] [32]. The methods outlined by [PERSON] and his followers are applicable not only to time series but also to random series, usually [24]. Figure 4: Fragment of the city plan (**a**) and a fragment of the Hilbert curve (**b**). While Considering the service area of the network, in which network elements (users or network nodes) are distributed in some random way (Figure 3), the number of elements found in the zone fragment will also be a random value. If the entire zone is divided into elementary fragments and the number of network elements is counted in them, then we get a random series on which the above methods are applicable. This approach forms the basis of the further construction of the model. In this study, the Hurst parameter \(H\) will be used to describe the network structure, which can reflect the manifestation degree of self-similarity properties as well as can be associated with the fractal dimension as \(D=2-H\)[24]. Let us estimate it for the structure shown in Figure 3 using the variance change method [33]. The process is the following. The service area under consideration will be divided into unit elements of the area, for example, by implementing a grid of square cells with an area of 1 m\({}^{2}\) (cells may even have a different area). First, let's count the number of hosts in each cell. As a result, we get a value of \(k\):  
\[X=\{x_{1},x_{2},\ldots,x_{k}\} \tag{4}\]  
where \(k\) is the number of grid cells. Next, we will construct the aggregated structures generated from the original (Figure 3) by combining the cells of the original partition with \(q\) adjacent cells into one. Then, we get \(k\) cells for the aggregated structure, where \(k=\frac{n}{q}\). Afterward, we count the number of network elements (nodes) in the aggregated partition cells and normalize them in comparison to the original partition. An illustration of the construction of aggregated structures is shown in Figure 6.  
Figure 5: Structure model based on a fragment of the (**a**) Hilbert curve, (**b**) Network elements. In the general case, for each partition \(i\), we will obtain a set of values:  
\[X_{i}=\{\frac{x_{1}}{q_{i}^{2}},\frac{x_{2}}{q_{i}^{2}},\ldots,\frac{x_{j}}{q_{i} ^{2}},\ldots,\frac{x_{k}}{q_{i}^{2}}\},k=(\frac{w}{q_{i}})^{2},i=1\ldots n \tag{5}\]  
where \(i\) is the number of the partition \(i=1\ldots n\), \(q_{i}\) is the size of the cell side into which the service area is divided \((m)\), \(x_{j}^{q}\) is the number of network elements (nodes) that fall into the \(j\)-th cell, size \(q_{i}\), \(w\) is the length of the side of the service area \((m)\), \(n\) is the number of partitions. For each value \(q_{i}\), we calculate the variance of the obtained values \(d_{i}=d(X_{i})\):  
\[D_{i}=\{d_{1},d_{2},\ldots,d_{n}\},k=(\frac{w}{q_{i}})^{2},i=1\ldots n \tag{6}\]  
The cell size take values from the set \(\{q_{1},q_{2},\ldots,q_{n}\}\).