# Arithmetic Study about Efficiency in Network Topologies for Data Centers
## 2 Topology Designs for Data Centers
### Graph-like Design  
Some instances of graph-like designs are going to be considered herein. The first design is the \(N\)-hypercube, where two nodes share each available line in a given dimension. This way, each node has \(N\) links to its neighbors: just one per dimension. The overall number of nodes is \(2^{N}\), and the distance between opposite nodes is \(N\). Figure 5 shows the nodes within \(N\)-hypercubes of lower dimensions. The second design is a folded \(N\)-hypercube, where the previous topology is taken and, in turn, links between each pair of opposite nodes are added. That way, each node has \(N+1\) links to its neighbors, and the distance between opposite nodes is just one. This implies that performance is improved although the design has become more complex. Figure 6 exposes the nodes within the folded \(N\)-hypercubes of lower dimensions. The third design is \(N\)-simplex, which is basically a full mesh topology, as all nodes are directly connected to each other. Hence, there are \(N+1\) nodes, where each of them has \(N\) links to its neighbors, resulting in a distance of one between any pair of nodes. Figure 7 exhibits the nodes for the \(N\)-simplices of lower dimensions. Figure 4: Devices in each layer in a redundant hub and spoke topologywith 6 spokes and a single hub (**left**) or a redundant hub (**right**). Figure 5: Nodes in \(N\)-hypercubes of dimensions \(\{0\cdots 4\}\) (from left to right). Figure 6: Nodes in folded \(N\)-hypercubes of dimensions \(\{2\cdots 4\}\) (from left to right). The fourth design is \(N\)-orthoplex, where the previous topology is taken and, in turn, links between each pair of opposite nodes are deleted, resulting in a quasi full mesh topology. Hence, there are \(2N\) nodes, whereby each of them has \(2(N-1)\) links to its neighbors, resulting in a distance of one between any pair of nodes except for opposite nodes, for which the distance is two. Figure 8 depicts the nodes for the \(N\)-orthoplices of lower dimensions. The fifth design is \(k\)-ary \(n\)-cube, also known as camcube, which is a toroidal topology. This is basically a grid where nodes at the edges of a certain line in a given dimension have a wraparound link, thus turning those into direct neighbors. The number of nodes is \(k^{n}\), and each node has \(2n\) links. Figure 9 exposes a couple of examples, where \(n\) accounts for the number of dimensions involved and \(k\) denotes the number of nodes within a given dimension. It is to be noted that if \(k=2\), then the shape obtained is that of the \(N\)-hypercube. The sixth design is Hamming graph \(H(n,n-1)\), whose topology may be seen as a folded version of the \(k\)-ary \(n\)-cube because an extra link directly connects each pair of nodes that are at the maximum distance, which in the previous case was \(\lfloor\
icefrac{{k}}{{2}}\rfloor\) when \(n=1\) and \(2\times\lfloor\
icefrac{{k}}{{2}}\rfloor\) if \(n=2\), whilst now it is half of those values, although complexity has grown in the design. Figure 10 exhibits an \(H(n,n-1)\) where \(n=2\) and \(n=4\). The seventh design is a Petersen graph, which is an instance of a cage graph. It contains \(10\) nodes connected in a way so that the maximum distance between nodes is two. It is also known as a \((3,5)\)-cage, as each node has a degree of three, meaning that all of them have three links towards other nodes, whilst the minimum loop available in the design contains five hops. Figure 11 shows the node disposition in a Petersen graph, wherein two concentric circles are spotted; i.e., the exterior oneis connected as a regular pentagon and the interior oneis linked as a pentagonal star. Figure 8: Nodes in \(N\)-orthoplices of dimensions \(\{1\cdots 4\}\) (from left to right). Figure 10: Nodes in Hamming graph \(H(n,n-1)\) (left: \(n=2\); right: \(n=4\)). Figure 9: Nodes in \(k\)-ary \(n\)-cube (left: 4-ary 1-cube; right: 4-ary 2-cube). The eighth design is a Heawood graph, which is also a cage graph. It contains 14 nodes interconnected in a way so that the maximum distance among nodes is three.