# On the Evaluation of Complex Networks Designs for an Energy-Efficient IP/WDM Core Network
## 4 Implementation of Small World, Random, and Scale-Free Networks on the NSF Network  
In this section, we implement the small-world network, ER random network, and SFN on the NSF to produce different topologies using methods and codes presented in [11] to further study the impact of such networks on the average path diameter, average clustering coefficient, and the total network power consumption. The NSF network as illustrated in Figure 2 consists of 14 nodes distributed in different states of the United States (US) with 21 bi-directional links. The network is referred to as the current NSF, and it has an average shortest path diameter of 2.175 and an average clustering coefficient of 0.0714. To create SFN graphs to interconnect the 14 nodes of the current NSF network illustrated in Figure 2, the algorithm described by [PERSON] and [PERSON] in [11] is implemented. Multiple runs of the algorithm generate different graphs and yield different ways of connecting the NSF 14 nodes with 21 edges. Each modeled complex network graph was run more than 10 times, and the samples provided in this work are our best graphs in terms of power consumption conservation. However, the mathematical MILP model for each sample has given the same results (optimum), regardless of the number of times the model was run. Samples of such generated SFN topologies are illustrated in Figure 3. Vertices in red represent the nodes with dense connectivity (hubs). To create small-world and random graphs to interconnect the 14 nodes of the current NSF network, the methods described by [PERSON] and [PERSON] in [11; 12] are implemented. Figures 4 and 5 illustrate samples of generated small-world and random graphs that mimic the NSF's 14-node connectivity model. The diameter of a network is the length of the shortest path between the farthest-distanced nodes. The cost metric is used to indicate which path is the shortest between all pairs of nodes. A cost measure of a graph can be the total distance, or the number of nodes traversed between every possible pair of source and destination nodes. Average path length is calculated by summing up all the shortest path lengths of all the source and destination pairs of nodes and dividing them by the total number of pairs. Figure 2: Base network topology: Current topology of the NSF network with time zones [4]. From the results illustrated in Figure 6, the current NSF network shows a relatively longer average shortest path when compared with the small-world network topology implemented in the NSF. It is observed from the results illustrated in Figure 6 that as the rewiring probability is increased, more shortcuts between nodes are increased, and as a result the path length decreases. A small fraction change in the rewiring probability creates shortcuts which can connect widely separated nodes, and hence results in a significant reduction in the diameter of the whole network [6]. Results illustrated in Figure 6 demonstrate that a reduction of 15% in the NSF network path length can be achieved if a small-world network design with \(p=1\) (as in Figure 4d) is considered for interconnecting nodes in the NSF network. Along with the shortest average path, small-world networks with \(p=1\) have a high node clustering coefficient. The clustering coefficient indicates how good a node is connected to its neighbors' neighbors. A high clustering coefficient indicates that the network comprises of many strong ties. Table 1 defines and demonstrates how the node's clustering coefficient can be calculated. If the node's clustering coefficient is 1, the neighbors of that node are fully connected, and when the neighbors are poorly connected, the value of the clustering coefficient will be near 0 [6]. Results illustrated in Figure 7 demonstrate a 246% increase in the clustering coefficient when compared with the current NSF design. Figure 3: Samples of SFN graph’s implementation on the NSF’s 14 nodes with 21 edges; (**a**) SFN sample 1, (**b**) SFN network sample 2, (**c**) SFN network sample 3.