# Arithmetic Study about Efficiency in Network Topologies for Data Centers
## 6 Discussion about the Results Obtained  
The study carried out above, from the point of view of arithmetic, presents some conclusions to be taken into account. To start with, the values of \(\eta\) obtained for tree-like designs are higher than those for graph-like designs when it comes to \(\alpha\), although regarding \(\beta\) it is the other way around. In other words, the average number of hops is greater for tree-like instances because of the excess of links due to the switching hierarchy, whilst this hierarchical nature allows nodes to have no interconnection with their peers, as those are undertaken among switches. Further, the different topologies proposed for a given number of hosts result in higher values of \(\eta\) as the number of redundant paths grows, which happens for both tree-like and graph-like designs. Focusing on the former, the redundant hub and spoke presents the lowest value, followed by leaf and spine, and finally, fat tree. Centering on the latter, partial mesh topologies attain lower values of \(\eta\), although the values grow as the number of links rises, increasing to maximum values for full-mesh topologies. Additionally, in tree-like designs, the average number of links among hosts increases with redundancy, whilst the average number of links per device decreases, resulting in \(\eta\) going higher, which shows that the weight of \(\alpha\) is bigger than \(\beta\) in tree-like topologies. On the contrary, in graph-like designs, the average number of links among hosts decreases with redundancy, whereas the average number of links per device increases, resulting in \(\eta\) going higher, which points out that the weight of \(\beta\) is bigger than \(\alpha\) in graph-like topologies. Having said that, further research could add some correction factor in order to try to balance the opposite relationship between \(\alpha\) and \(\beta\) for both kinds of architectures when using the proposed expression of \(\eta\). On the other hand, the values of performance and simplicity achieved with the other commonly used network topologies typically employed in data centers, described in Section 2.3 and calculated in Section 5.7, are significantly lower than their counterparts analyzed in the previous subsections within Section 5 for similar or even higher numbers of nodes. The reason of this is because the interconnections among nodes are optimized related to fat tree or leaf and spine, thus not presenting a pure tree structure butdisplaying  
\begin{table}
\begin{tabular}{l l c c c} \hline \hline
**Type** & **Instance** & **Nodes** &  
Figure 28: Coefficient for topologies with 8 nodes. Figure 30: Coefficient for small topologies not being a power of 2. Figure 29: Coefficient for topologies with 16 nodes. In summary, by reviewing the results obtained in Tables 1-5 for the coefficient proposed herein, namely \(\eta\), it may be seen that the values attained for both tree-like and graph-like cases increase as the number of nodes rises. Further, the results shown in Table 6 are reasonable comparing the values achieved with those shown in the previous tables according to the number of nodes, as they are all within the same range. Regarding the other commonly used data center network topologies presented in Table 7, the results are also coherent according to the amount of nodes included in each topology compared to those seen in the aforesaid tables. It is to be taken into account that Table 7 represents topologies where the number of nodes cover a wide range of values, as each topology has its own specifications. Nonetheless, most of the results obtained for \(\eta\) are located in the lower range in the previous tables despite Table 7 instances having a larger number of nodes in many cases. Therefore, it seems that this coefficient \(\eta\) offers significant results for data center network topologies focused onsmall deployments. However, as exposed in previous sections, this coefficient is just another tool to select the most convenient data center network design, not the definitive one.