1 Introduction

Nowadays, there is no doubt that the development of widely understood technologies related to communication networks is one of the most vital tasks the scientific community has to face. Without communication networks it would not be possible to develop any distributed technologies. So, it is well-timed to try to create a network with maximum efficiency and optimal and economical use of transmission resources. The performance of a communication network depends on how its connection topology is designed. These networks can be modelled using graphs, so graph theory plays an important role here. There are many network topologies that have been studied for their application to network modelling, such as ring, mesh, torus and hypercube. Each topology has some properties specific to it. One network topology suitable for telecommunication networks is the chordal ring, proposed by Arden and Lee in 1981 [1]. A chordal ring is a homogeneous, undirected cyclic graph. An important property of this type of network is the degree of nodes, understood as the number of edges of a graph incident to its vertex. It is an intermediate structure between a ring and a full graph, as it combines the properties of both these topologies. Due to the larger number of paths connecting the source node to the destination node, it is more resilient than a ring, while having a smaller diameter and average path length. The use of the chordal ring topology in large networks shows an advantage over full graphs, whose parameters mentioned above have lower values, but are characterised by an exponential increase in the number of connections as the number of nodes increases, which in turn is associated with their higher, economically unjustifiable cost. Since the implementation of the chordal ring topology was proposed in the early 1980s, many researchers have focused on improving and proposing new versions of this topology. This was due to its favourable parameters such as low latency in the transmission of information, fault tolerance and symmetry. The initial proposal to use third degree chordal rings based on a fixed chord length has consistently been developed and improved. One such improvement has been to increase the degree of classical topologies, for example to the fourth degree [2]; fifth degree [3] and even sixth degree [4]. Another way to further improve the aforementioned favourable parameters was to change the ways in which the chordal ring was connected, resulting in different types of nodes in the classes. Examples of such modified chordal rings were the modified sixth-degree ring presented in papers [5, 6]. The properties of the proposed chordal ring topologies such as symmetry [7] and Hamiltonicity and asymmetry [8] have also been studied over the years. Researchers are also interested in topics related to the application of these topologies in compact routing [9], optimal free-table routing [10] and broadcasting [11, 12]. New proposals to generate chordal ring structures continue to emerge, among others [13]. These constructs find applications in increasingly modern telecommunication technologies, including wireless networks [14], sensor networks [15]. Graph theory is used in solving problems that can be represented by graphs, such as algorithms used in network analysis or to describe phenomena related to social engineering, technology, computer engineering [16] and many other complex real-world systems [17]. As mentioned, a variety of problems can be described and analyzed using chordal ring structures, which makes them a useful tool. Graph theory can be said to be a versatile tool for describing and solving problems involving relationships between objects in various scientific fields.

2 Basic Information and Concepts

As it was proved in the work [18], transmission properties of networks that are described by graphs depend not only on their diameter and average path length, but also on the use of particular edges of these structures for data transmission, which is best illustrated by the example of Reference Graphs [19, 20], which, despite having the same underlying parameters, may differ in the properties mentioned above.

Definition 1

Reference Graphs are regular structures with a predetermined number of nodes in which the diameter values and the average path lengths from any source node reach the same, theoretically calculated lower size limits.

Figure 1 shows examples of such graphs of the fourth degree formed by seven nodes with diameters of 2 and average path lengths of 1.333.

Fig. 1
figure 1

Analysed reference graphs

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These structures were subjected to simulation tests and the chart (Fig. 2) shows the results. To carry out the tests, simulation software developed by the authors of this article and described in detail in the publication [21] was used. As a parameter determining the transmission properties of the network, the probability of rejecting a service call in the function of changes in the traffic generated in the network nodes was chosen. In this case, the tests were carried out under the assumption that 64 transmission units are assigned to each edge and that the same amount of traffic is generated at each node.

Fig. 2
figure 2

Simulation results. T: traffic volume generated in nodes measured in Erlangs, Prej: probability of rejecting a service call

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As already mentioned, the factor causing differences in the transmission properties of the networks described by these graphs is the uneven use of individual edges of these structures, the measure of which is a parameter called the unevenness coefficient [22].

Definition 2

The unevenness coefficient \(w_{spi}\) is the parameter determining the number of uses of a given edge in sets of parallel paths (routes of the same length created by various configurations of these edges) connecting vertices of a graph.Individual edges can be a part of multiple paths, even those that connect the same nodes.

An unevenness coefficient is described by the formula:

$$\begin{aligned} w_{spi}=\sum _{i=1}^{D(G)} u_{io} \end{aligned}$$
(1)

where D(G) is the diameter of the graph and \(u_{io}\) values calculated the formula:

$$\begin{aligned} u_{io}=\frac{u_{k}}{k} \end{aligned}$$
(2)

\(u_{k}\) means the number of uses of a particular edge in the sets of parallel paths of count k [23]. In the case under consideration (Fig. 1), the calculated \(w_{spi}\) coefficients have the values given in Table 1 (e i denotes the edge number of the graph).

Table 1 Determined coefficient \(w_{spi}\) values
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The resource control principle is implemented according to the values of the calculated imbalance coefficients, which allows more efficient management of the global transmission resources of the analysed network. It involves allocating more resources to edges that are used more frequently to transmit information, at the expense of reducing them for edges carrying less traffic. The size of network resources allocated to each edge is calculated using the rule:

$$\begin{aligned} RES_{i}=d_{i}\cdot RES_{g},\quad where \quad d_{i}=\frac{w_{spi}}{\sum w_{spi}} \end{aligned}$$
(3)

where \(RES_{i}\) is the resources used by the \(i-th\) edge, \(RES_{g}\) is the total network resources, and \(\sum w_{spi}\) is a sum of the coefficients calculated for all edges [24]. In the case under consideration \(RES_{g}\) = 896 contractual units, while in both cases the value of \(\sum w_{spi}\) is 56. Table 2 shows the distribution of resources for the individual edges of the graphs.

Table 2 Distribution of resources allocated to individual edges of graphs
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After adjusting the transmission resources, simulation studies were carried out again and the result is shown in Fig. 3.

Fig. 3
figure 3

Simulation results after resource adjustment. T: traffic volume generated in nodes measured in Erlangs, Prej: probability of rejecting a service call

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Based on the results obtained, it can be concluded that by introducing a network resource control that uses the results of the calculation of the unevenness coefficients, an alignment of the characteristics of the Reference Graphs with equal number and degree of nodes can be achieved, while the size of the global resources does not change.

3 Equalization of Transmission Characteristics of Networks Described by Regular Graphs with Different Diameters and Average Path Lengths

The reference graphs discussed in the previous part of the article are characterised by the same basic parameters, while, as it was specified in the introduction, the analysis and research aimed to check whether it is possible to equalise the characteristics of networks described by graphs whose basic parameters differ [25]. The following example is used to explain the principle and the procedure. Simulation tests of networks described by chord rings of degree three with 14 nodes (Fig. 4) were carried out, and the graph (Fig. 5) shows the obtained results assuming that 64 units are assigned to each edge.

Fig. 4
figure 4

The graphs describing the studied networks\(. D(G)_{A} = 3, d_{avA} = 2.0769; D(G)_{B} = 4, d_{avB} = 2.3846\)

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Fig. 5
figure 5

Results of simulation tests of networks described by 14-node third-degree graphs A, B (Fig. 4). T: traffic volume generated in nodes measured in Erlangs, Prej: probability of rejecting a service call

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It was assumed that to equalize the transmission characteristics of the studied graphs, it is sufficient to calculate the ratio of their average path lengths and multiply the resources corresponding to the edges of the graph with the larger average path length by a ratio factor called Ref, which is calculated from the formula:

$$\begin{aligned} Ref_{G1/G2}=\frac{d_{avG1}}{d_{avG2}} \end{aligned}$$
(4)

\(d_{avG1}\) is the average path length of the graph under the study, \(d_{avG2}\) is the average path length of the reference graph.

The adjusted values of the resources, which should make it possible to achieve similar transmission properties of both networks described by the above-mentioned graphs, were calculated using the following formula:

$$\begin{aligned} RES_{icG1}=Ref_{G1/G2}\cdot RES_{iG1} \end{aligned}$$
(5)

In the analysed example, the value of \(Ref_{B/A}\) for graph A is 1.148153. After taking this value into account, the edges of graph B were assigned 73 units, which means that the global resources would increase by 189 units. The graph (Fig. 6) illustrates the obtained results of the tests.

Fig. 6
figure 6

The results of simulation tests of networks described by 14-node graphs of the third degree after the modification of the resources of graph B resulting from the inclusion of \(Ref_{dav}\) values T: traffic volume generated in nodes measured in Erlangs, Prej: probability of rejecting a service call

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The accompanying diagram shows that this way of using the \(Ref_{G1/G2}\) coefficient does not guarantee an equalisation of the transmission properties of the tested networks. In the search for a method that would result in achieving the assumed objective, i.e., similar transmission characteristics of the network, the method used for aligning the transmission properties of Reference Graphs was applied.

Using the determined values of the unevenness coefficients given in Table 3, the resource values for the individual edges of graph B were adjusted (Table 4), which, as mentioned earlier, results in more efficient use of the transmission properties of the network. Sum of unevenness coefficients \(\sum w_{spiB} = 434\). In the case of graph A, the values of all unevenness coefficients are equal to 18 and therefore the resources assigned to its edges have not changed and are equal to 64 units and \(\sum w_{spiA} = 378\). The results for graph B described above are shown in Fig. 7.

Table 3 The determined values of the parameters \(w_{spi}\) (e\(_i\) stands for the edge number)
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Table 4 Adjusted and rounded resource values for individual edges of graph B
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Fig. 7
figure 7

Results of the study of analysed graphs after resource modification. Bc denotes the test results of graph B after introducing the correction resulting from the calculated values of the \(w_{spi}\) parameters. T: traffic volume generated in nodes measured in Erlangs, Prej: probability of rejecting a service call

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Adjusted values of \(RES_{icc}\) resources have been calculated, which should make it possible to obtain similar transmission properties of the networks described by the above-mentioned graphs:

$$\begin{aligned} RES_{icc}=Ref_{G1/G2}\cdot RES_{ic} \end{aligned}$$
(6)

Table 5 gives the results of the calculations of the resources assigned to the edges of the graphs after considering the values of the \(Ref_{B/A}\) parameter

Table 5 Results of resource calculations after taking \(Ref_{B/A}\) into account
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The chart (Fig. 8) shows the simulation results.

Fig. 8
figure 8

Results of simulation tests after the adjustment of transmission resources taking into account the value of the \(REF_{B/A}\) parameter. T: traffic volume generated in nodes measured in Erlangs, Prej: probability of rejecting a service call

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To achieve the assumed goal of the undertaken activities, i.e., to equalise the characteristics of data transmission in networks described by graphs with different basic parameters (diameter, average path length), it is necessary to increase the total transmission resources for graph B by 203 units. Further, to verify that the minimum value of resources necessary to obtain a probability Prej close to the reference graph was achieved, simulation tests were carried out on the network described by graph B, with changes in the value of the parameter \(Ref_{B/A}\) and in the volume of generated traffic (30–70) [Erl]. The results of the calculations are presented in Table 6.

Table 6 Prej values in the function of \(Ref_{B}\) parameter changes
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Figure 9 shows the differences in the values of the probability of obtaining correct transmission in relation to graph A.

Fig. 9
figure 9

Variations in the probability values of rejecting a service call \(\Delta \) Prej with respect to the characteristic values for graph A, as a function of changes in the generated traffic T measured in Erlangs

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It has been checked whether the proposed mode is also effective when the networks are described by graphs with different average path lengths but the same diameter. For this purpose, the graphs shown in Fig. 10 were examined.

Fig. 10
figure 10

Analysed graphs with twenty-three nodes

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The diameter of both graphs is equal to 4, while the average path lengths are \(d_{avA}\) = 2.3636 and \(d_{avB}\) = 2.4545. Figure 11 shows the comparative results of the simulation tests without the introduction of corrections to the network resources assigned to individual edges (each edge was assigned 64 units), while Fig. 12 shows the results of the tests after the introduction of corrections to the resources, also taking into account the impact of the value of the determined parameter \(Ref_{B}\) equal to 1.0385.

Fig. 11
figure 11

Test results of the networks described by the graphs shown in Fig. 10. T: traffic volume generated in nodes measured in Erlangs, Prej: probability of rejecting a service call

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Fig. 12
figure 12

Network test results: Ac and Bc: after the initial resource adjustment, Bcc: after taking into account the value of the \(Ref_{B}\) parameter. T: the intensity of traffic generated in nodes measured in Erlangs, Prej: the probability of rejecting a service call

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Based on the obtained results, it can be concluded that the average length of the paths plays a decisive role in the alignment of the characteristics of the studied networks, while the size of the diameter of the graphs describing these networks is virtually irrelevant, of course under the condition of equal number and degree of nodes describing these networks.

4 Equalization of Transmission Characteristics of Networks Described by Regular Graphs with More Complex Structure

This part of the article describes the results of the examination of structures that are closer in their topology to real conditions, but while maintaining the regularity condition. Figure 13 shows an example of a virtual network described by a third-degree regular graph with 26 nodes.

Fig. 13
figure 13

Tested network structure described by a regular graph of the third degree

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The values of the basic parameters of the graph shown in Fig. 13 are as follows: the diameter is 7 and the average path length is 3.6954.

The network described by the chord graph shown in Fig. 14 was chosen as a reference structure, with a diameter of 5 and an average path length of \(d_{av}\) = 2.84.

Fig. 14
figure 14

Reference graph of the third degree with 26 nodes

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Figure 15 shows the results of simulation tests performed assuming that each edge (there are 39 of them) is assigned 64 units of transmission resources, which means that the global transmission resources of these two networks are 2496 units.

Fig. 15
figure 15

Test results of the simulation networks described by the graphs shown in Figs. 15a and 16b. T: traffic volume generated in nodes measured in Erlangs, Prej: probability of rejecting a service call

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It was checked what results would be obtained by applying the \(Ref_{A}\) factor to correct the resources assigned to the edges. The value of this coefficient is 1.3012, so each edge of the graph is assigned 83 transmission units. Figure 16 illustrates the obtained results of the simulation.

Fig. 16
figure 16

Results of the simulation tests of the networks described by the analysed graphs after taking into account the value of the \(Ref_{A}\) parameter (Aref). T: traffic volume generated in nodes measured in Erlangs, Prej: probability of rejecting a service call

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As can be seen from the graph, the obtained transmission characteristics of network A improved slightly, but deviated from those of network B, so a method using unevenness coefficients was used. Their values for graph A have been calculated and are shown in Table 7. All \(w_{spi}\) values of the graph B are the same and equal 47.333.

Table 7 Unevenness coefficient values
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To make more rational use of the network capacity, the sizes of the network resources allocated to each edge were calculated (Table 8) using the results in Table 7.

Table 8 Set of network resources assigned to the edges of a graph (after rounding)
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Fig. 17
figure 17

Simulation results after modifying the resources of graphA(Ac). T: traffic volume generated in nodes measured in Erlangs, Prej: probability of rejecting a service call

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The values of the resources assigned to the edges of graph B, due to equal \(w_{spi}\) values, have not changed, hence they are 64 units. The original modification of the resources assigned to the edges of graph A was made, and the simulation tests were repeated, the results of which are shown in Fig. 17.

In this case a sum of the coefficients calculated for all edges \(\sum w_{spiA}= 2402, \sum w_{spiB}= 1846\). Using the formula (5), the adjusted resource values were calculated taking into account the value of the parameter \(Ref_{A}= 1.3012\). (Table 9).

Table 9 Resources assigned to the edges of graph A after taking into account the \(Ref_{A}\) parameter value
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The results of the simulations carried out are shown in Fig. 18 (\(RES_{icc}\) values are rounded to integer values).

Fig. 18
figure 18

Simulation results after modifying the resources taking into consideration the value of the Ref(Acc) parameter. T: traffic volume generated in nodes measured in Erlangs, Prej: probability of rejecting a service call

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By introducing the resource control, similar transmission characteristics of both networks were obtained after increasing the overall transmission resources of graph A by 724 units. Additional tests were carried out by modifying the allocated network resources in a function of changes in the \(Ref_{A}\) parameter. Table 10 presents the calculation results obtained and Fig. 19 shows the corresponding simulation results.

Table 10 Prej values with changes in the generated traffic in the range of 15–35 [Erl] at different values of the parameter Ref
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Fig. 19
figure 19

Variations in the probability values of rejecting a service call \(\Delta \) Prej for the characteristic values for graph A, as a function of changes in the generated traffic T measured in Erlangs

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5 Summary and Conclusions

Based on the analysis of the obtained results it has been stated that at the cost of increasing the transmission resources assigned to particular network edges, calculated with the use of the Ref parameter, it is possible to obtain similarity of transmission properties of networks described by regular graphs of equal degree and with the same number of nodes but different diameter sizes and average path lengths, without the necessity of changing the topology of internodal connections and with minimal increase in the network resources used for information transmission.

The result of the considerations contained in the paper is the following algorithm, aimed at equalizing the transmission properties of networks described by graphs with the same number of nodes and the same number of nodes of a given degree.

  1. 1.

    Preliminary operations:

    • Selecting from the analysed networks the one described by a graph with shorter average path length.

    • Running a preliminary simulation to compare the transmission properties of the network assuming equal resources for each edge.

    • Calculation of unevenness coefficients for both studied networks.

    • Calculation of the value of transmission resources after an adjustment to make more rational use of these resources.

    • Performing a simulation after the adjustment.

  2. 2.

    Equalisation of transmission characteristics:

    • Determining the aggregate coefficient wspi values.

    • Calculation of the RefG1/G2 coefficient.

    • Adjusting the network resources taking into account the RefG1/G2 value.

    • Performing tests after the adjustment.

    • Determination of modification costs (size of additional transmission units).