# Edge Data Center Organization and Optimization by Using Cage Graphs
## 1 Introduction  
Data center deployments are exponentially growing because of the rise in edge and fog domains powered by AI-based technologies [1]. This situation is bringing more attention to data center organization and optimization [2] in order to improve their performance in IoT environments [3]. In this context, machine learning techniques may help optimize performance by achieving increased efficiency in resource usage [4], leading to a decrease in energy consumption [5], which may lower the carbon footprint [6] in order to make the IoT environments more sustainable [7] and resilient [8].  
Modern data centers are composed of physical hosts where remote computing resources are allocated within [9], usually being virtual machines or virtual containers [10]. In this sense, communications among virtual resources within the same physical machine are internal, which are much faster than east-to-west traffic, which takes place among diverse physical devices. The former takes place among internal means within a physical host, whereas the latter occurs through network connections among physical hosts [11].  
Hence, in order to improve the performance of data centers, it is convenient to optimize the allocation strategy of virtual resources within physical nodes in order to gather interdependent resources and minimize the number of hops among nodes [12], which may lead to shorter migration times of the virtual resources between any pair of physical devices [13]. Such strategies to obtain efficient resource utilization are usually implemented through the use of generic algorithms, where multiple factors need to be taken into consideration, such as the current and predicted state of the virtual resources, the current and predicted application workloads, or the cost of migration to other hosts [14], as well as the data center network topology.  
In this paper, a data center layout is proposed in order to minimize the number of hops between any pair of physical hosts, thus reducing the migration cost. The topology proposed is based on graph theory and, more specifically, on cage graphs, where all nodes share the same number of neighbors and the shortest cycle within the design. Furthermore,some instances of small cage graphs are presented, such as Petersen, Heawood, and Robertson graphs, which may be useful for small to medium data centers, as its network diameter is only two in the first or three in the second and third, wehreas the number of links is not as high as that of complete graphs.  
Additionally, we developed an algorithm to forward traffic through the 10 nodes within a Petersen graph in an optimized manner, thus providing a method to quickly reach any other node by using integer divisions and modular arithmetic, as opposed to searching for a match throughout the traditional routing tables or mac-address lists. The algorithmic method needs fewer operations to find the proper link to send traffic toward its destination, even though its actual performance improvement rate may depend on how it is implemented in hardware. Additionally, another algorithm devoted to Heawood graph was designed to forward traffic among its 14 nodes where the key point is the distance between source and destination node, as well as the outline to carry out a further algorithm for the Robertson graph to move among its 19 nodes.  
The rest of this paper is organized as follows: Section 2 shows the motivation of this study. Section 3 presents an overview of graph theory regarding integrity and cage graphs; Section 4 explains the advantages of a Petersen graph to boost performance in small data centers, which is followed by Section 5, constructing a Petersen graph for small data centers. Section 6 is devoted to Heawood graphs and Robertson graphs; then, Section 7 presents an experimental analysis comparing the aforementioned topologies with other commonly used in data centers. Finally, Section 8 draws some final conclusions.