# Arithmetic Study about Efficiency in Network Topologies for Data Centers
## 1 Introduction
It is estimated that power distribution consumes around 15% of the total energy consumption, whilst cooling systems use around 45%, thus leaving the remaining approximately 40% to the IT equipment, which is shared between networking equipment (taking between 30% and 50%, depending on the load level) and computing servers (taking the rest of it) [28]. In this sense, it is to be noted that some part of that energy is consumed in over-provisioning of resources to meet requirements during demand at peak times [29]. Hence, it is crucial to undertake an appropriate network design to optimize the overall performance [30]. The approach to achieve efficiencyin this paper is set on data center topology. Some strategies have been presented in the literature, such as employing deep learning tools [31] or focusing on wireless environments [32]. However, the strategy proposed herein is about network topology: that being the logical topologythat links together all nodes belonging toa data center [33] and its influence in the overall performance [34]. In this sense, a specific coefficient is going to be defined so as to combine the average number of hops to reach any destination withina data center architecture and the average number of links per node in that architecture, which is known as the degree in graph theory. This way, the former stands for a measure of performance, whereas the latter does it for a measure of simplicity, thus obtaining a metric that showsa trade-off between performance and simplicity, similar to the reasoning behind the metric proposed in [35]. It is to be noted that the concept of performance taken in this paper is related to communications among the nodes within the data center, which is usually measured in time units. However, if it is considered a data center where all nodes are interconnected with ethernet cables that have the same length and the same speed, then performance may also be measured in distance units, as speed is the result of dividing distance by time, which accounts for distance as the result of multiplying speed by time. Hence, as speed is a constant value because all cable links have the same featuresin this case, then performance measured in distance units is directly proportional to that measured in time units, as the measure given in distance units equals the speed of the wire multiplied by the measure given in time units. In other words, performance in distance units equals to that quoted in time units multiplied by a constant of proportionality, which happens todepend on the wirespeed thus, there is a proportional relationship between both ways of expressing performance. Furthermore, as all cabling has the same length, then it is possible to easily convert performance given in distance units into the minimum number of cables to be traversed from a source node all the way to a destination node by just dividing the former by the length of each cable given in distance units. This way, the measure of performance in distance units may also be exposed as the number of links among nodes, which is expressed in natural numbers as it is an adimensional quantity. Therefore, this is the reason why the average number of hops between any pair of nodes is considered as a measure of performance of a certain data center topology in this paper. Additionally, in order to clarify concepts in this context, the number of hops between a given pair of nodes is also called the distance between them both, with the result being that the shorter the distance, the better the performance. In other words, the shorter the average number of hops between any pair of nodes, the better the performance of the data center network topology interconnecting those nodes. With respect to the concept of simplicity, it has been considered as a measurement of the ease of a topology to manage the routing and forwarding processes, as the smaller the number of links in a device, the simpler the algorithm to handle traffic is, as there are fewer possible cases to be evaluated.