# 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 sixth design is SlimFly, which approaches the theoretically optimal network diameter, thus shrinking the average distance among nodes [47]. This topology is achieved through mathematical optimization looking for the Moore bound in graph theory, reducing more than 50% of routers and 30% of cables with respect to fat tree. It is fit for use in large data centers and high-performance computing (HPC). It obtains a 25% cost and power benefit over DragonFly as well as being more resilient to link failures. Figure 19 represents a Slim Fly topology where two subgraphs are composed of five groups of routers, and those groups form a fully connected bipartite graph such that each group in one subgraph is connected to all other groups in the other subgraph [48]. The seventh design is BCDC, which is a server-centric data center network topology based on crossed cubes [49]. This is a decentralized and recursively defined structure where servers have a constant degree, which possesses an advantage over DCell or BCube in this sense whilst getting better results than them. Figure 20 exposes a three-dimensional BCDC topology where switches are shown as squares and hosts are circles and with all switches having three ports and all servers having two. Figure 19: Nodes in a SlimFly topology, with groups of 5 routers forming a fully connected bipartite graph of diameter 2. Figure 20: Nodes in a BCDC topology in 3 dimensions. The eighth design is P-Cube, also labeled as parallel cubes; it is a duplicate structure with a highly scalable and efficient network structure that outperforms Fat Tree, BCube and DCell in terms of network throughput and latency [50]. Figure 21 shows a P-Cube network topology where \(n=4\) and switches are represented by squares and servers are shown as circles. The ninth design is DCCube, which is a compound graph with the disc-ring graph and the crossed cube CQ\({}_{n}\)[51]. It supports a large number of nodes and has high bisection bandwidth, low cost and a small diameter. Figure 22 shows an instance with one dimension (\(k=1\)), four switches per pod (\(m=4\)), one port per switch pointing to other pods (\(h=1\)) and two servers connected per switch (\(c=2\)), although many other combinations may be undertaken. It is to be noted that black solid lines represent links within a given pod, whilst blue dotted lines show links among pods, whereas red solid lines indicate the boundaries of each pod. The tenth design is Jellyfish network, which adopts a random regular graph (RRG) as its topology [52] and has been provento outperform fat tree [53]. Jellyfish may be specified with three parameters, such as the number of switches (\(N\)), the number of ports in each switch (\(x\)) and the number of ports in each switch connecting to other switches (\(y\)), thus resulting in \(x-y\) nodes connected to each switch. It happens that when \(N\) and \(y\) are large enough, different instances have similar features. Figure 23 exhibits a Jellyfish instance with \(N=15\), \(x=4\) and \(y=3\), where switches are denoted as squares and nodes are shown as circles. It is to be said that switches and nodes have been separately identified in a random manner within the picture, starting from zero onwards  
Figure 21: Nodes in a P-Cube topology with \(n=4\). Figure 12:0 Nodes in a DCCube topology with \(k=1\), \(m=4\), \(h=1\) and \(c=2\). The eleventh design presents Subways, which is a novel approach based on wiring servers to Top-of-Rack (ToR) switches located in neighboring racks [54]. This way, instead of having all links connected to the same ToR, an overlapping pattern is used for neighboring racks. The advantages of doing so are a decrease in traffic in the inter-ToR network whilst ensuring that the remaining traffic is well-balanced. Figure 24 exhibits a cluster with a Type-\(1\) subway architecture with \(p=3\) and \(l=3\), where the former is the number of ports per server and the latter is the number of racks in a single Subway loop.