# Arithmetic Study about Efficiency in Network Topologies for Data Centers
## 2 Topology Designs for Data Centers
### Other Commonly Used Network Topologies in Data Centers  
The following network topologies are typically implemented in large data centers, even though the instances exhibited in this subsection have the parameters set for a small  
Figure 11: Nodes in Petersen graph, a \((3,5)\)-cage with 10 nodes. Figure 12: Nodes in Heawood graph, a \((3,6)\)-cage with 14 nodes. Figure 13: Nodes in Robertson graph, a \((4,5)\)-cage with 19 nodes. to medium number of servers. Moreover, it is to be noted that all the following designs are tree-like, as they all present a hierarchical structure. The first design is BCube, which is a recursively defined structure specially designed for modular data centers [42]. Regarding its construction, it contains nodes with multiple ports and switches connecting a constant number of nodes. To start with, a BCube\({}_{0}\) contains just \(n\) nodes connected via an \(n\)-port switch. Furthermore, a BCube\({}_{1}\) is built up from \(n\) BCube\({}_{0}\) with \(n\)\(n\)-port switches and so on. This way, a BCube\({}_{k}\)\((k\geq 1)\) consists of \(n\) BCube\({}_{k-1}\) and \(n^{k}\)\(n\)-port switches, with each node having \(k+1\) ports. Figure 14 exhibits the node layout in a BCube\({}_{1}\) with \(n=4\). The second design is DCell, which is also a recursively defined structure specifically designed for modular data centers [43]. The building block for its construction is DCell\({}_{0}\), which has \(n\) nodes and a miniswitch. Further, DCell\({}_{1}\) is made of \(n+1\) DCell\({}_{0}\) blocks, where each of those is connected to the other DCell\({}_{0}\) blocks with just one link. Figure 15 depicts the node layout in a DCell\({}_{1}\) with \(n=4\). The third design is FiConn, which is a recursively defined structure as well and is properly designed for modular data centers [44]. It is to be noted that the node degree is always two, which makes interconnection easier than in DCell. The basic construction unit is FiConn\({}_{0}\), and it is composed of \(n\) nodes and \(n\)-port switches, where all \(n\) nodes have their backup port available. Then, to build up FiConn\({}_{k}\)\((k>0)\) upon FiConn\({}_{k-1}\), it is necessary to connect the backup ports of nodes among them. Figure 16 exposes the node layout in FiConn\({}_{2}\) with \(n=4\). Figure 14: Nodes in a BCube\({}_{1}\) topology with \(n=4\). Figure 15: Nodes in a DCell\({}_{1}\) topology with \(n=4\). The fourth design is Flattened Butterfly, whose main use is devoted to on-chip networks. Regarding data center network architectures, its main feature is the presence of bypass channels in both horizontal and vertical dimensions, propitiating the employment of non-minimal routing without increasing latency or energy consumption [45]. Figure 17 presents on the left-hand side a block of 16 switches distributed as a \(4\times 4\) layout both horizontally and vertically, where any two switches along the same line are direct neighbors thanks to the bypass channels. It is to be noted that each of those switches supports four nodes, although they are not shown in the picture for clarity purposes. Further, on the right-hand side, a block of nine switches distributed as a \(3\times 3\) layout is depicted, with three hosts hanging on each switch. The fifth design is DragonFly, which reduces cost by 20% compared to Flattened Butterfly and by 52% compared to folded Clos networks such as fat tree for more than 16K nodes [46]. Basically, it uses a group of routers with bypass channels among them that acts as a virtual router to increment the effective radix of the network, thus reducing the network diameter, cost and latency. Figure 18 depicts a Dragonfly topology with \(g=9\) groups and \(a=4\) routers within each group. Regarding the number of nodes per router, which is not shown in the picture for simplicity purposes, it should be \(p=a/2=2\) in order to balance the channel load. Figure 16: Nodes in FiConn\({}_{2}\) topology with \(n=4\). Figure 17: Nodes in a Flattened Butterfly topology with \(n=4\) (**left**) and \(n=3\) (**right**).