## 3 Case Studies
### From Survivable to Resilient Topologies: 2-Connected Graphs  
This second case study focused on survivability requirements. Information on the invariants values of the proposed topologies can be found in Table 3.  
At first sight, the search for resilient topology solutions for a network with 14 nodes could lead to the 2-connected ring topology shown in Figure 3a, which contains only 14 edges and has average degree 2 (low cost). However, as discussed in Section 2, the ring is a poor solution considering latency and resilience requirements, because the diameter is high (7, in this case) and changes to 13 in case of any node removal. Moreover, given a fixed number of nodes, the ring is the 2-connected graph maximizing the Wiener index [29]. For a ring with 14 nodes, the Wiener index is 343, and the node Wiener impact of each of its nodes is 70. With the goal of minimizing these impacts, an expert could propose the solution shown in Figure 3b [37], which attempts to combine the benefits from ring and star topologies. This solution reduces the latency related invariants (the average distance, from 3.77 to 1.71, and the diameter, from 7 to 2) and the Wiener index by adding 12 edges from each node to a central node. On the other hand, its benefits are still dependent on a single node, which is related to the high values for maximum  
Figure 2: Tree based topologies: (**a**) star; (**b**) expert; and (**c**) improved solution (AGX). degree (13) and degree variance (6.63). The maximum and minimum values for Wiener impact (130 and 0, respectively) also demonstrate the significance of removing this central node on the transmission of the other nodes and, thus, show that the solution is not resilient. Once again our solution supports the refinement of this topology to address its faults. Starting with the solution presented in Figure 3b and using AGX, we obtained the graph shown in Figure 3c. Here, the optimization minimizes the Wiener impact, and the average distance, subject to a 2-connected graph of both diameter and maximum degree less than or equal to four. The number of edges is also fixed to 26, which allows a fair comparison between both solutions with respect to interconnection parameters. The maximum degree constraint is an example requirement from [36]. Our optimized solution has few drawbacks: the average distance goes from 1.71 to 2.04, the diameter goes from 2 to 3, and the Wiener index goes from 156 to 186. On the other hand, it has great positive impact on the maximum degree (from 13 to 4) and the degree variance (from 6.63 to 0.20). Moreover, our method heavily improves the topology resilience as shown by the reduction on the maximum Wiener impact (from 130 to 6). Extending the previous analysis, we investigate C-RAN topologies, which are typically composed by hierarchical ring structures [38], as shown in Figure 4a. This structure is not a 2-connected graph and leads to poor resilience properties, since the removal of a single node that interconnects two rings produces a disconnected graph. We optimized the C-RAN topology presented in Figure 4a using AGX to improve resilience, and obtained the graph shown in Figure 4b. The graph optimization problem can be stated as: minimize the Wiener impact, and the average distance, subject to a 2-connected graph of diameter less than or equal to 6 and maximum degree less than or equal to four. Table 4 compares the hierarchical ring topology with the optimized one, which has much better resilience properties. As the topology based on hierarchical rings is not a 2-connected graph, it is not possible to calculate its Wiener impact (i.e., some nodes have infinite Wiener impact, since some  
Figure 3: 2-connected topologies: (**a**) ring; (**b**) wheel; and (**c**) improved solution (AGX).