## 3 Case Studies
### Non-Survivable Topologies: Trees  
This first case study focused on node interconnection and latency requirements. Table 2 collects the invariants values of the proposed topologies.  
In the search for topologies for a network with 19 nodes, basic interconnection constraints and a minimal latency objective could be addressed with a classic star topology (tree graph family), as shown in Figure 2a, which contains only 18 edges (low edge cost), has diameter 2 (low maximum latency), has average distance 1.89 (low average latency), and has Wiener index 324. Unfortunately, analysis showed that this topology is highly dependent on a central node (maximum degree is 18, and degree variance is 14.41), whose removal stops the communication between all other nodes. Moreover, the high maximum degree brings implementation barriers.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Invariants** & **(a) Star** & **(b) Expert [36]** & **(c) Improved Solution (AGX)** \\ \hline Max. degree & 18 & 4 & 4 \\ Average degree & 1.89 & 1.89 & 1.89 \\ Degree variance & 14.41 & 1.67 & 1.78 \\ Diameter & 2 & 6 & 5 \\ Average distance & 1.89 & 3.18 & 3.16 \\ Wiener index & 324 & 544 & 540 \\ Max./Min. edge betweenness & 18/18 & 70/18 & 78/18 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Tree based topologies (19 nodes).  
Figure 1: Topology design method.  
Although a high centralized topology would bring some impacts to the network interconnectivity, for C-RAN, such centralization would be perfect for reusing the resources at data-centers. The more centralized a BBU pool is positioned, the bigger is the potential to receive offloaded workload from other RANs to be processed in the same BBU pools. It means that the operator would not have to spend too much revenue with new BBU pools by reusing the already deployed ones.  
Despite the benefits that C-RAN would afford from such topology, to minimize the centralization impacts, an expert could propose the solution shown in Figure 2b [36]. This solution effectively reduces the maximum degree and the degree variance from 18 to 4 and from 14.41 to 1.67, respectively, with the downside of increasing the diameter from 2 to 6, and increasing the Wiener index from 324 to 544.  
Our method allows the further systematic refinement of even the expert's solution. From the solution presented in Figure 2b and using AGX, our approach obtains the graph shown in Figure 2c. The optimization applied here minimizes the diameter and the average distance, subject to maximum degree less than or equal to four. The maximum degree constraint ensures that the gains originally achieved by the expert are maintained. The result of this optimization only changes one edge to reduce the diameter from 6 to 5 while keeping almost all other invariants, with the only disadvantage of slightly increasing the load on some edges (maximum edge betweenness goes from 70 to 78, and degree variance goes from 1.67 to 1.78). Alternatively, we could consider different objective functions and constraints to improve other topological parameters. For example, if the constraint on the maximum degree is not so critical to the network design, we could use maximum degree less than or equal to five to reduce the diameter to 3. That is, our approach to topology design enables the systematic exploration of potential topologies and enumeration of major trade-offs.