# Arithmetic Study about Efficiency in Network Topologies for Data Centers
## 2 Topology Designs for Data Centers
### Tree-like Design  
Three instances of tree-like topologies are going to be taken into account herein. The first design is a fat tree, whose main feature is the establishment of three layers of switches, where the lower one is called the edge layer (which is the one in touch with the nodes), the middle one is named the aggregation layer, and the upper one is branded the core layer [39]. This design has strict specifications, such as it establishes a parameter \(k\) that governs the number of hosts and switches in each layer as well as the ports linking different layers.  
Figure 2 exhibits the devices in each layer in a fat tree topology where \(k=4\) and the oversubscription rate is 1:1, meaning that no expected links are missing. In that picture,  
Figure 1: Tree-like design (**left**) -vs- graph-like design (**right**).  
\(H\) represents the nodes (also called hosts), \(E\) represents the edge switches, \(A\) represents the aggregation ones, and \(C\) represents the core ones, where elements of the same kind are sequentially numerated from left to right. It is to be noted that a fat tree architecture organizes switches in the lower and middle layers in groups whereby full mesh connectivity among both layers within a single group is achieved. Those groups are called pods, and there is full mesh connectivity among them through the switches in the upper layer, although each individual switch in the middle layer just gets partial mesh connectivity with all upper-layer switches.  
The second design is leaf and spine, whose main characteristic is the establishment of two layers of switches, where the lower one is named the leaf and the upper one is called the spine [40]. In this case, no parameter governs the number of devices or ports among layers, so there is some freedom of design when it comes to choosing the number of hosts and switches in each layer. Figure 3 exposes the devices in each layer in a leaf and spine topology with eight leaf switches, these are represented by \(F\), and four spine switches, branded as \(G\), whilst the number of nodes per each leaf is not fixed. It is to be said that there is full mesh connectivity between switches located in both layers.  
The third design is hub and spoke, which may be considered two-level treestructure wherein the hub is on top of the hierarchy and the spokes are at the bottom [41]. In many production environments, it is quite common to use a redundant hub and spoke design, where two hubs are used for redundancy purposes and a number of spokes are connected to each of them, and where network traffic may be load-balanced so as to try to leverage all links. Figure 4 exhibits on the left-hand side the devices within each layer in a single hub and spoke topology with one switch acting as a hub and six nodes acting as spokes, whereas on the right-hand side, a redundant hub and spoke topology replicates the former scheme by adding another hub into the design for redundancy purposes. In both cases, all interspoke communications take place through a hub, so that every pair of spokes is just two hops away.  
Figure 3: Devices in each layer in a leaf and spine topologywith 8 leaves and 4 spines.  
Figure 2: Devices in each layer in a fat tree topologywith \(k=4\) and oversubscription rate 1:1.