## 4 Graph Theoretic Resiliency Framework
### Resiliency Estimation Framework  
Let us consider the network model using the graph approach as described earlier. Now, we consider the representation of the network by an undirected graph, as shown in Figure 2. We assume that there are two paths between the PMU and PDC such that the paths are not distinct, as a node is observed to be common in two paths. To be noted, except for the source and destination nodes, the other nodes between the PMUs and PDCs are not common. Thus, the paths between the PMU and PDC are distinct if the source and destination nodes are ignored.  
At the beginning, all the nodes in \(v\in V\) are in their normal states. Furthermore, all links \(e_{i-j}\in E\) connecting nodes \(v_{i}\in V\) and \(v_{j}\in V\), such that \(\{i,j\}=\{1,2,3,\ldots,\eta|i\
eq j\}\), are operational. Consequently, the PMU and PDC will be able to exchange the data as the network is operational. Now, consider at time \(t_{1}\), link \(e_{1-6}\in E\) partially fails. The failure of this node results in the necessity to reroute the data through another possible path. There will be degradation in the \(PDR\) beyond time \(t_{1}\), and this degradation depends upon the route recovery time (RRT). As node \(v_{2}\) is a shared node having more background traffic than the previous path, the \(PDR\) is always less than 1 but greater than threshold value of this path. The \(PDR\) can be improved such that \(PDR\) becomes close to 1 if, and only if, the failed link \(e_{1-6}\in E\) is restored to its normal state. This can be seen beyond time \(t_{4}\) in Figure 4.  
If the background traffic over link \(e_{2-3}\in E\) is assumed to be large enough, then the \(PDR\) degrades in such a way that \(PDR\) becomes less than the threshold value. This results in the disruption of service. The main cause of the disrupted service is the failed state of the link \(e_{1-6}\in E\) with no backup path, such as through \(e_{2-3}\in E\).  
Initially, we considered all nodes to be operational. Thus, they participated in the data exchange, resulting in \(PDR=1\). For simplicity, all nodes and links are assumed to be of equal capacity. Link \(e_{2-3}\in E\) works as a shared link between the \(v_{1}-v_{6}\) and \(v_{0}-v_{6}\) source-destination pairs. Thus, for equal sharing, half of the link capacity will be used for each of the source-destination pairs, whereas the remaining half will be considered as the background traffic. Thus, for each source-sink pair, there exist two paths: \(\gamma_{1-6}^{1}=\left\{\,v_{1},v_{6}\right\}\) and \(\gamma_{1-6}^{2}=\left\{\,v_{1},v_{2},v_{3},v_{4},v_{5},v_{6}\right\}\). Here, the dominant path for the \(v_{1}-v_{6}\) source-destination pair becomes \(\gamma_{1-6}^{1}=\left\{\,v_{1},v_{6}\right\}\). We assume that the dominant path carries 80% of the \(PDR\) data and the other paths carry 20%. It is worth mentioning that the choice of data carried by the dominant path is optional and subject to the researcher's choice. The exact path between the PMU and PDC for the routing of the data can be estimated by any of the suitable network layer-routing techniques, which is not part of this work. In fact, the routing protocol will be used for the simulation of the network using ns-3, which will be discussed in the later section of proposed work.  
Next, the network's resiliency will be estimated under four cases: _normal state_, _partially failed state_, _restoration state_, and _fully failed state_.