#
Evaluation of Transmission Properties of Networks Described with Reference Graphs Using Unevenness Coefficients^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

- The minimum network connection cost presented as the total number of links;
- The minimum communication delay—the representation of this parameter is the size of the diameter and the average path length;
- A substantial fault tolerance characterized by the number of independent paths between two nodes (connectivity) or the minimum number of nodes or edges after the removal of which the networks is no longer consistent (node and edge connectivity);
- Regularity and symmetry;
- Ease of routing;
- Extensibility.

**Definition**

**1.**

**Definition**

**2.**

## 2. The Adopted Method of Analyzing the Topic

## 3. The Method of Proceeding

**Conclusion:**The value of the parameter ${w}_{spi}$ determines the number of occurrences of a given edge in the minimum length paths.

- The diameter of each of the structures D(G)4 = 2; average path length ${d}_{av}$ = 1.5.
- The first layer, as well as the second one, consists of four nodes; thus, ${d}_{sum}=4+4\xb72=12$ edges.
- RGs have the same parameter values calculated from any node, so the global number of edges forming the minimum paths is $\sum {d}_{sum}=9\xb712=108$.

**Conclusion:**In Reference Graphs with an identical number and degree of nodes, the total length of all minimum length paths $\sum {d}_{sum}$ is equal to the total value of all ${w}_{spi}$ coefficients. Using the obtained results shown in Table 7, the authors calculated and analyzed the standard deviation $\sigma $ of the studied coefficients from the average value ${w}_{spi\phantom{\rule{0.222222em}{0ex}}av}$. The deviation is calculated according the following formula:

**Conclusion:**Based on the analysis of the determined values of $\sigma $, it is possible to compare the transmission properties of networks described by graphs without performing simulation tests. The smaller the value of $\sigma $, the better the transmission properties of the network described by the RG graph. However, it is not a measure of these properties but merely an indicator. The analysis of the results showed that for the selected number of nodes constituting Reference Graphs of a given degree, the total number of coefficients of unevenness is strictly defined. Its exemplary values are given in Table 14 and Table 15.

## 4. Summary and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Results of simulating the probability of rejecting a call in the function of traffic density for graphs A and B (Figure 1). ${P}_{rej}$—probability of rejecting the call for realization, T—density of the generated traffic measured in Erlangs.

**Figure 6.**Results of simulations for graph 41 compared to graph 122: (

**A**) without the correction procedure; (

**B**) with the correction procedure.

Graph A | |||||||||

Node | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

0 | be | ae | bh + cj | dn | af + cm | ag + dr | bi + ds | cn | |

1 | be | ab | eh + fk | ac + fm | gp | fp | ei + gq | ad + gr | |

2 | ae | ab | il | bc + hj | ef + hk | eg + iq | hl | bd + is | |

3 | bh + cj | eh + fk | il | km | jm | kp + lq | hi | jn + ls | |

4 | dn | ac + fm | bc + hj | km | jk | mp + nr | jl + ns | cd | |

5 | af + cm | gp | ef + hk | jm | jk | fg | kl + pq | mn + pr | |

6 | ag + dr | fp | eg + iq | kp + lq | mp + nr | fg | rs | qs | |

7 | bi + ds | ei + gq | hl | hi | jl + ns | kl + pq | rs | qr | |

8 | cn | ad + gr | bd + is | jn + ls | cd | mn + pr | qs | qr | |

Graph B | |||||||||

Node | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

0 | be | ae + ch | bh | af + bi + cj | dp | ag + dr | ck + ds | 0 | |

1 | be | ab + fi | ac + eh + fj | ei | fl + gm | 0 | gq | ad + gr | |

2 | ae | ab + fi | bc + ij | ef + hj | il | eh | hk | bd | |

3 | bh | ac + eh + fj | bc + ij | hi | jl + kn | kq | 0 | cd + ks | |

4 | af + bi + cj | ei | ef + hj | hi | 0 | fg + ml | jk + ln | lp | |

5 | dp | fl + gm | il | jl + kn | 0 | nq + pr | mq + ps | mr + ns | |

6 | ag + dr | 0 | eg | kq | fg + lm | nq + pr | mn + rs | mp + qs | |

7 | ck + ds | gq | hk | 0 | jk + ln | mq + ps | mn + rs | np + qr | |

8 | 0 | ad + gr | bd | cd + ks | lp | mr + ns | mp + qs | np + qr |

Node | Path Length | Edge | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

a | b | c | d | e | f | g | h | i | j | k | l | m | n | p | q | r | s | ||

A | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |

2 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |

∑ | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | |

B | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |

2 | 8 | 12 | 4 | 8 | 4 | 8 | 12 | 4 | 4 | 4 | 4 | 4 | 8 | 12 | 12 | 4 | 4 | 4 | |

∑ | 10 | 14 | 6 | 10 | 6 | 10 | 14 | 6 | 6 | 6 | 6 | 6 | 10 | 14 | 14 | 6 | 6 | 6 |

Graph A | ||||||||

∑ | ${\mathit{u}}_{\mathit{is}}$ | ∑ | ${\mathit{u}}_{\mathit{is}}$ | ∑ | ${\mathit{u}}_{\mathit{is}}$ | |||

a | 10 | 749,768 | g | 10 | 750,701 | m | 10 | 749,352 |

b | 10 | 749,772 | h | 10 | 749,435 | n | 10 | 749,864 |

c | 10 | 751,121 | i | 10 | 749,537 | p | 10 | 749,560 |

d | 10 | 750,356 | j | 10 | 750,510 | q | 10 | 750,118 |

e | 10 | 750,146 | k | 10 | 750,839 | r | 10 | 748,635 |

f | 10 | 751,249 | l | 10 | 749,668 | s | 10 | 749,801 |

Graph B | ||||||||

∑ | ${\mathit{u}}_{\mathit{is}}$ | ∑ | ${\mathit{u}}_{\mathit{is}}$ | ∑ | ${\mathit{u}}_{\mathit{is}}$ | |||

a | 10 | 666,682 | g | 14 | 1,249,334 | m | 10 | 666,768 |

b | 14 | 1,248,994 | h | 6 | 499,130 | n | 14 | 1,251,003 |

c | 6 | 583,250 | i | 6 | 500,822 | p | 14 | 1,251,017 |

d | 10 | 667,232 | j | 6 | 749,417 | q | 6 | 749,858 |

e | 6 | 584,238 | k | 6 | 583,680 | r | 6 | 499,736 |

f | 10 | 665,674 | l | 6 | 582,844 | s | 6 | 500,906 |

Graph A | |||||||||

Node | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

0 | 0 | a | b | bh + cj | c | af + cm | ag + dr | bi + ds | d |

1 | a | 0 | e | eh + fk | ac + fm | f | g | ei + gq | ad + gr |

2 | b | e | 0 | h | bc + hj | ef + hk | eg + iq | i | bd + is |

3 | bh + cj | eh + fk | h | 0 | j | k | kp + lq | l | jn + ls |

4 | c | ac + fm | bc + hj | j | 0 | m | mp + nr | jl + ns | n |

5 | af + cm | f | ef + hk | k | m | 0 | p | kl + pq | mn + pr |

6 | ag + dr | g | eg + iq | kp + lq | mp + nr | p | 0 | q | r |

7 | bi + ds | ei + gq | i | l | jl + ns | kl + pq | q | 0 | s |

8 | d | ad + gr | bd + is | jn + ls | n | mn + pr | r | s | 0 |

Graph B | |||||||||

Node | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

0 | 0 | a | b | c | af + bi + cj | dp | ag + dr | ck + ds | d |

1 | a | 0 | e | ac + eh + fj | f | fl + gm | g | gq | ad + gr |

2 | b | e | 0 | h | i | il | eg | hk | bd |

3 | c | ac + eh + fj | h | 0 | j | jl + kn | kq | k | cd + ks |

4 | af + bi + cj | f | i | j | 0 | l | fg + lm | jk + ln | lp |

5 | dp | fl + gm | il | jl + kn | l | 0 | m | n | p |

6 | ag + dr | g | eg | kq | fg + lm | m | 0 | q | r |

7 | ck + ds | gq | hk | k | jk + ln | n | q | 0 | s |

8 | d | ad + gr | bd | cd + ks | lp | p | r | s | 0 |

Edge | k | ${\mathit{w}}_{\mathit{spi}}$ | Edge | k | ${\mathit{w}}_{\mathit{spi}}$ | Edge | k | ${\mathit{w}}_{\mathit{spi}}$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | ||||||

a | 2 | 4 | 4 | 5.33 | g | 6 | 8 | 0 | 10.00 | m | 2 | 4 | 4 | 5.33 |

b | 6 | 8 | 0 | 10.00 | h | 2 | 4 | 0 | 4.00 | n | 6 | 8 | 0 | 10.00 |

c | 4 | 0 | 2 | 4.67 | i | 2 | 4 | 0 | 4.00 | p | 6 | 8 | 0 | 10.00 |

d | 2 | 4 | 4 | 5.33 | j | 6 | 0 | 0 | 6.00 | q | 6 | 0 | 0 | 6.00 |

e | 6 | 8 | 0 | 4.67 | k | 4 | 0 | 2 | 4.67 | r | 2 | 4 | 0 | 4.00 |

f | 2 | 4 | 4 | 5.33 | l | 4 | 0 | 2 | 4.67 | s | 2 | 4 | 0 | 4.00 |

Edge | ${\mathit{u}}_{\mathit{ci}}$ | ${\mathit{u}}_{\mathit{si}}$ | Edge | ${\mathit{u}}_{\mathit{ci}}$ | ${\mathit{u}}_{\mathit{si}}$ | Edge | ${\mathit{u}}_{\mathit{ci}}$ | ${\mathit{u}}_{\mathit{si}}$ |
---|---|---|---|---|---|---|---|---|

a | 666,666.7 | 666,682 | g | 1,250,000.0 | 1,249,334 | m | 666,666.7 | 666,768 |

b | 1,250,000.0 | 1,248,994 | h | 500,000.0 | 499,130 | n | 1,250,000.0 | 1,251,003 |

c | 583,333.3 | 583,250 | i | 500,000.0 | 500,822 | p | 1,250,000.0 | 1,251,017 |

d | 666,666.7 | 667,232 | j | 750,000.0 | 749,417 | q | 750,000.0 | 749,858 |

e | 583,333.3 | 584,238 | k | 583,333.3 | 583,680 | r | 500,000.0 | 499,736 |

f | 666,666.7 | 665,674 | l | 583,333.3 | 582,844 | s | 500,000.0 | 500,906 |

**Table 7.**Values of ${w}_{spi}$ coefficient obtained by simulation for the different fourth-degree RGs.

RG Number | 122 | 17 | 27 | 28 | 5 | 25 | 4 | 2 | 1 | 13 | 6 | 32 | 23 | 10 | 29 | 41 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${w}_{spi}$ | 6 | 4.67 | 4.67 | 5.33 | 4 | 4 | 5 | 4 | 3.67 | 4 | 3.67 | 2 | 2 | 4 | 4 | 2 |

6 | 4.67 | 4.67 | 5.33 | 4 | 4 5 | 5 | 3.67 | 4 | 3.67 | 2 | 3.67 | 4 | 4 | 2 | 2 | |

6 | 5.67 | 4.67 | 5.33 | 5.67 | 5 | 5 | 5 | 5 | 4 | 4 | 5.5 | 3.67 | 4 | 4 | 4 | |

6 | 5.67 | 6 | 5.33 | 5.67 | 5 | 5 | 5.17 | 5 | 4 | 5 | 5.5 | 4 | 4 | 4 | 4 | |

6 | 5.67 | 6 | 5.33 | 5.67 | 5.33 | 5 | 5.17 | 5 | 4 | 5 | 5.5 | 5.33 | 4 | 4.67 | 5 | |

6 | 5.67 | 6 | 5.33 | 5.67 | 5.33 | 5 | 5.17 | 5 | 4 | 5.33 | 5.5 | 5.33 | 4 | 4.67 | 5 | |

6 | 6.17 | 6 | 5.33 | 5.67 | 5.67 | 5 | 5.17 | 5.33 | 7 | 5.67 | 5.5 | 5.67 | 4 | 4.67 | 5 | |

6 | 6.17 | 6 | 5.33 | 5.67 | 5.67 | 5 | 5.67 | 5.33 | 7 | 5.67 | 5.5 | 5.67 | 4 | 4.67 | 5 | |

6 | 6.17 | 6 | 5.33 | 5.67 | 5.67 | 6 | 5.67 | 5.67 | 7 | 5.67 | 5.5 | 6.67 | 4 | 5.33 | 5 | |

6 | 6.17 | 6 | 6.67 | 5.67 | 5.67 | 6 | 5.67 | 5.67 | 7 | 5.67 | 5.5 | 6.67 | 8 | 5.33 | 5 | |

6 | 6.17 | 6 | 6.67 | 6 | 6.33 | 6 | 5.67 | 6.67 | 7 | 6 | 7 | 6.67 | 8 | 5.33 | 5 | |

6 | 6.17 | 6 | 6.67 | 6 | 6.33 | 6 | 6 | 7 | 7 | 6 | 7 | 6.67 | 8 | 5.33 | 5 | |

6 | 6.17 | 6.67 | 6.67 | 6.67 | 6.67 | 6 | 6.83 | 7 | 7 | 7 | 7 | 6.67 | 8 | 6 | 8 | |

6 | 6.17 | 6.67 | 6.67 | 6.67 | 6.67 | 6 | 6.83 | 7 | 7 | 7 | 7 | 6.67 | 8 | 6 | 8 | |

6 | 6.67 | 6.67 | 6.67 | 6.67 | 7.33 | 8 | 6.83 | 7 | 7 | 7.67 | 8 | 7.33 | 8 | 10 | 8 | |

6 | 6.67 | 6.67 | 6.67 | 6.67 | 7.33 | 8 | 6.83 | 7.33 | 7 | 7.67 | 8 | 7.33 | 8 | 10 | 8 | |

6 | 6.67 | 6.67 | 6.67 | 8 | 8 | 8 | 8.67 | 7.33 | 7 | 8.67 | 8 | 9 | 8 | 10 | 12 | |

6 | 6.67 | 6.67 | 6.67 | 8 | 8 | 8 | 8.67 | 9.33 | 7 | 8.67 | 8 | 9 | 8 | 10 | 12 | |

$sum$ | 108 |

RGNumber | 122 | 17 | 27 | 28 | 5 | 25 | 4 | 2 |

$\sigma $ | 0 | 0.577 | 0.667 | 0.667 | 1.018 | 1.155 | 1.155 | 1.207 |

RGNumber | 1 | 13 | 6 | 32 | 23 | 10 | 29 | 41 |

$\sigma $ | 1.395 | 1.414 | 1.483 | 1.732 | 1.767 | 2.000 | 2.222 | 2.749 |

Edge | a | b | c | d | e | f | g | h | i |

${\mathit{w}}_{\mathit{spi}}$ | 12 | 5 | 4 | 5 | 5 | 4 | 5 | 5 | 2 |

Edge | j | k | l | m | n | p | q | r | s |

${\mathit{w}}_{\mathit{spi}}$ | 8 | 5 | 12 | 8 | 8 | 8 | 5 | 2 | 5 |

**Table 10.**Auxiliary Table 1.

Edge | a | b | c | d | e | f | g | h | i |

$\Delta {\mathit{w}}_{\mathit{spi}}$ | 6 | −1 | −2 | −1 | −1 | −2 | −1 | −1 | −4 |

Edge | j | k | l | m | n | p | q | r | s |

$\Delta {\mathit{w}}_{\mathit{spi}}$ | 2 | −1 | 6 | 2 | 2 | 2 | −1 | −4 | −1 |

**Table 11.**Auxiliary Table 2.

Edge | a | b | c | d | e | f | g | h | i |

$\Delta \mathit{RES}$ | 32.00 | −5.33 | −10.67 | −5.33 | −5.33 | −10.67 | −5.33 | −5.33 | −21.33 |

Edge | j | k | l | m | n | p | q | r | s |

$\Delta \mathit{RES}$ | 10.67 | −5.33 | 32.00 | 10.67 | 10.67 | 10.67 | −5.33 | −21.33 | −5.33 |

Edge | a | b | c | d | e | f | g | h | i |

$\Delta \mathit{RES}$ | 64 | 27 | 21 | 27 | 27 | 21 | 27 | 27 | 11 |

Edge | j | k | l | m | n | p | q | r | s |

$\Delta \mathbf{RES}$ | 43 | 27 | 64 | 43 | 43 | 43 | 27 | 11 | 27 |

Graph | a | b | c | d | e | f | g | h | i |

27 | 5.333 | 5.333 | 6.667 | 6.667 | 5.333 | 6.667 | 6.667 | 6.667 | 6.667 |

28 | 4.667 | 6.000 | 6.667 | 6.667 | 6.000 | 6.667 | 6.667 | 6.000 | 6.000 |

Graph | j | k | l | m | n | p | q | r | s |

27 | 5.333 | 6.667 | 5.333 | 5.333 | 6.667 | 5.333 | 5.333 | 6.667 | 5.333 |

28 | 6.667 | 4.667 | 6.000 | 4.667 | 6.667 | 6.000 | 6.000 | 6.000 | 6.000 |

NodeNumber | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 | 26 | 28 | 30 |

$\sum {w}_{spiN}$ | 42 | 88 | 150 | 252 | 378 | 528 | 702 | 900 | 1122 | 1416 | 1742 | 2100 | 2490 |

D(G) = 2 | D(G) = 3 | D(G) = 4 |

NodeNumber | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 |

$\sum {w}_{spiN}$ | 36 | 56 | 80 | 108 | 140 | 176 | 216 | 260 | 308 | 360 | 416 | 476 |

D(G) = 2 | ||||||||||||

NodeNumber | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 |

$\sum {w}_{spiN}$ | 558 | 646 | 740 | 840 | 946 | 1058 | 1176 | 1300 | 1430 | 1566 | 1708 | 1856 |

D(G) = 3 |

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**MDPI and ACS Style**

Bujnowski, S.; Marciniak, B.; Lutowski, Z.; Flizikowski, A.; Oyerinde, O.O.
Evaluation of Transmission Properties of Networks Described with Reference Graphs Using Unevenness Coefficients. *Electronics* **2021**, *10*, 1684.
https://doi.org/10.3390/electronics10141684

**AMA Style**

Bujnowski S, Marciniak B, Lutowski Z, Flizikowski A, Oyerinde OO.
Evaluation of Transmission Properties of Networks Described with Reference Graphs Using Unevenness Coefficients. *Electronics*. 2021; 10(14):1684.
https://doi.org/10.3390/electronics10141684

**Chicago/Turabian Style**

Bujnowski, Sławomir, Beata Marciniak, Zbigniew Lutowski, Adam Flizikowski, and Olutayo Oyeyemi Oyerinde.
2021. "Evaluation of Transmission Properties of Networks Described with Reference Graphs Using Unevenness Coefficients" *Electronics* 10, no. 14: 1684.
https://doi.org/10.3390/electronics10141684