# Edge Data Center Organization and Optimization by Using Cage Graphs
## 7 Experimental Analysis
### Comparing Experimental Analysis with Other Common Topologies  
In order to experimentally analyze the proposed cage graphs, some common statistic measurements were made, referring to both central tendency and dispersion, as well as the appropriate coefficient of variation. These statistics refer to the number of hops from a given node within a topology to any other one, along with the number of links per node in each topology. In this way, the former may be associated with to the performance of the system: the lower the number of hops to reach any other peer, the better. Otherwise, the latter may be assigned to ease of operation and maintenance of the topology: the lower the amount, the simpler the interconnections and packet forwarding. Ten topologies were selected for this statistic analysis, where full mesh was the ideal one in theory, although it does not scale well in practice. Moreover, two instances of \(N\) hypercubes were taken, such as those where \(N=\{3,4\}\), along with their two folded \(N\)-hypercube counterparts. Additionally, two instances of \(k\)-ary \(n\) cube were chosen: 3-ary 2-cube and 4-ary 2-cube. Eventually, the three instances of small cage graphs studied herein were also considered: Petersen, Heawood, and Robertson graphs. Those commonly used topologies provide a suitable representation of simple architectures for small data centers. To start, Table 7 presents the average number of hops to reach any other node and the average number of links per node. The average number of hops away, also known as the arithmetic mean values, was calculated by summing the multiplication of the different numbers of hops from one node to the others by the number of nodes at that distance, all divided by the number of nodes to reach overall, where the reference node is not taken into consideration. The average numbers of links per node were easily obtained, as the topologies studied herein are all graph-like designs, where all nodes have the same number of links; hence, the average is just that number with no dispersion. Focusing on the average number of hops, other statistic measurements are listed in Table 8. The mode and median were calculated regarding central tendency measurements, whereas variance and standard deviation represent dispersion measurements. We then calculated the coefficient of variation for all those topologies, which is the standard deviation divided by the arithmetic mean. It results in a dimensionless value between 0 and 1, where values above 0.30 are not considered to be homogeneous, meaning that the arithmetic mean is not representative of the data set [42]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Topology** & **Average Number** & **Average Number** \\  & **of Hops** & **of Links Per Node** \\ \hline Full mesh (8 nodes) & (1-7)/7 = 1 & 7 \\
3-hypercube (8 nodes) & (1-3+2-3+3-1)/7 = 1.71 & 3 \\
4-hypercube (16 nodes) & (1-4+2-6+3-4+1)/15 = 2.13 & 4 \\ Folded 3-hypercube (8 nodes) & (1-4+2-3)/7 = 1.43 & 4 \\ Folded 4-hypercube (16 nodes) & (1-5+2-10)/15 = 1.67 & 5 \\
3-ary 2-cube (9 nodes) & (1-4+2-4)/8 = 1.50 & 4 \\
4-ary 2-cube (16 nodes) & (1-4+2-6+3-5)/15 = 2.13 & 4 \\ Petersen graph (10 nodes) & (1-3+2-6)/9 = 1.67 & 3 \\ Heawood graph (14 nodes) & (1-3+2-6+3-4)/13 = 2.07 & 3 \\ Robertson graph (19 nodes) & (1-4+2-12+3-2)/18 = 1.89 & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Average number of hops and average number of links per node. Discussion  
From the point of view of the average number of links per node, the most repeated value is four links per item. Thus, higher values result in more complex topologies from the point of view of operation and maintenance, even though performance improves, which is the case of folded four-hypercube compared with their counterparts of 16 nodes. The case of full mesh beats all of the rest performance-wise. Some topologies present three links per node, such as the three-hypercube, along with the Petersen and Heawood graph designs.