## 3 Case Studies
### From Survivable to Resilient Topologies: A Real-World Network  
In C-RAN, its underlying infrastructure presents two problems: (i) massive initial investment; and (ii) ossified infrastructure. In the former, BBU pools need to be connected to RRHs using redundant optical links (e.g., ring links) that require site installation and long length links (e.g., \(\geq\)10 km) per antenna, hindering the centralization of BBU pools due to high cost. In the latter, the underlying infrastructure after installed cannot be easily repositioned needing new investments and man-working, compromising the scalability of C-RAN. Based on these problems, reusing already deployed underlying infrastructure (backhaul) enables the cut of initial investments also giving new possibilities of deployment, merging C-RAN's underlying infrastructure with backhaul. Although the reuse of the backhaul is crucial to the realization of C-RAN, the survivability of such hybrid underlying infrastructure may be compromised, mainly because of the different already deployed infrastructure purposes, such as interconnecting cities or providing connectivity to Internet Service Providers. Therefore, the goal of this case study is to extend our resilience analysis to a \begin{table}
\begin{tabular}{c c c} \hline \hline
**Invariants** & **(a) Hierarchical Ring [38]** & **(b) Improved Solution (AGX)** \\ \hline Nr. of edges & 16 & 22 \\ Max. degree & 4 & 4 \\ Average degree & 2.46 & 3.38 \\ Degree variance & 0.71 & 0.70 \\ Diameter & 6 & 6 \\ Average distance & 2.72 & 2.49 \\ Wiener index & 212 & 194 \\ Max./Min. edge betweenness & 33.5/6.5 & 11/6.25 \\ Max./Min. node Wiener impact & 0/\(\infty\) & 0/0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: CRAN topologies (13 nodes). Figure 4: CRAN topologies: (**a**) hierarchical ring; and (**b**) improved solution (AGX). real-world backbone network in the realization of C-RAN. To this end, we optimize the Brazilian National Research and Educational Network (RNP) topology (Available at www.pop-sc.rnp.br/servicos/conectividade), which is composed by 28 points of presence. Since the resilience analysis only makes sense in a 2-connected graph, Figure 5 shows the representation of the resultant 25-node topology after the removal of the three nodes that are not part of any cycle. Information on the invariants values of the proposed topologies can be found in Table 5.  
An analysis of Figure 5 shows that the highlighted nodes have high Wiener impact values (181 and 146), which tell us that the impact on the overall network resilience of removing these nodes is high. This diagnosis can drive an expert to connect extra edges in the network to reduce the impact of their disconnection. For example, in the graph of Figure 6a, the addition of only two extra edges (highlighted in red dashed lines) to the original graph of Figure 5 reduces the maximum Wiener impact from 181 to 69. This example shows how the analysis of node invariants can be used as an incremental and informed method for increasing network resilience in an existing network that does not support disruptive changes. Figure 5: RNP backbone: resultant 2-connected graph with 25 nodes. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Invariants** & **Original** & **Addition of Two Edges** & **Minimization of Max. WI** \\ \hline Nr. of edges & 38 & 40 & 38 \\ Max. degree & 7 & 7 & 5 \\ Average degree & 3.04 & 3.20 & 3.04 \\ Degree variance & 2.12 & 2.48 & 0.68 \\ Diameter & 5 & 5 & 5 \\ Average distance & 2.84 & 2.74 & 2.76 \\ Wiener index & 852 & 822 & 828 \\ Max./Min. edge betweenness & 55.17/6.50 & 45.85/6.23 & 32.74/12.33 \\ Max./Min. node Wiener impact & 181/0 & 69/0 & 34/0 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: RNP backbone (25 nodes). On the other hand, if the goal is to design a new topology from scratch that focuses on resilience, we could fix the original number of nodes (25) and edges (38) and try to minimize the maximum Wiener impact.