# Quantitative Characterization of Complex Systems--An Information Theoretic Approach
## 5 Case Study--Entropy Metric Formulation and Analysis of Simulated Complex Networks
### Simulation Models Used to Apply the Metrics
#### 5.3.2 Small World Networks  
The network generated here using NetLogo simulation software is based on the phenomenon of small word networks. This phenomenon implies that a given person is only a small number of (a few) connections away from any other person in the world. A popular example of this phenomenon is the famous [PERSON] network (also known as a six-degree separation network), which relates to a network generated based on actors appearing in the same movie. However, small world networks are not only limited to  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Time-Step** & **Entropy** & \(\mathbf{R_{0}}\) \\ \hline
0 & 1.00 & 0.00 \\
1 & 1.50 & 0.083 \\
2 & 1.79 & 0.083 \\
3 & 2.00 & 0.075 \\
4 & 2.32 & 0.046 \\
5 & 2.58 & 0.031 \\
6 & 2.80  
networks of people, but also apply to several other real-time networks such as power grids. The network model generated here is based on a few assumptions and conditions under which a small world network is formed. It is developed based on the model suggested by [PERSON] and [PERSON] [37]. The model starts by initially generating a network in which each node is connected to its two neighbors on either of its sides [35]. After this initial random network is generated, at each and every time-step, a random connection is picked and then rewired i.e., a random end of a connected pair of nodes is changed. The probability that a random edge is chosen and rewired is based on the rewiring probability value assigned to the network generation model. Figure 6 illustrates the evolution of interactions in the network over fifteen different snapshots. Tables 5 and 6 illustrate the node degree evolution observed for the generated small world network and their respective transfer entropy values. As observed in Figure 7, Node 0 has no information originating from it. This simply reflects the rigidity of the node, which does not change the degree over the network emergence (the node is connected to the same neighbors throughout), and thereby does not generate any new information. Figure 8 and Table 7 illustrate the entropy measure and R\({}_{0}\) calculated at every time-step of the dynamic network. It is observed that, as the network evolves, entropy decreases, implying that the small world network tries to reach an ordered state from a highly disordered initial state. As entropy (diversity) reduces the reproduction rate R\({}_{0}\) increases. This implies that when this network tends to reach absolute order (i.e., less entropy) the spread of information increases over the network. \begin{table}
\begin{tabular}{c c c c} \hline
**Source Node** & **Destination Node** & **Transfer Entropy (S-D)** & **Transfer Entropy (D-S)** \\ \hline
0 & 1 & 0.000 & 0.0002 \\
1 & 2 & 0.069 & 0.0006 \\
2 & 3 & 0.045 & 0.0882 \\
3 & 4 & 0.206 & 0.0106 \\
4 & 5 & 0.022 & 0.0137 \\
5 & 6 & 0.010 & 0.0059 \\
6 & 7 & 0.006 & 0.0185 \\
7 & 8 & 0.012 & 0.0077 \\
8 & 9 & 0.021 & 0.0609 \\
9 & 10 & 0.000 & 0.0000 \\
10 & 11 & 0.000 & 0.0022 \\
11 & 12 & 0.084 & 0.1661 \\
12 & 13 & 0.052 & 0.0267 \\
13 & 14 & 0.028 & 0.0226 \\ \hline \end{tabular}
\end{table} 
Table 6: TE values calculated based on node degrees from Table 5.  
Figure 7: Transfer entropy values between the nodes of small world network. The framework of information theory used has no restrictions on the type of complex system it is applied to because the interactions among the components/subsystems of a complex system can be viewed as information transfer. Perceived applications of the proposed framework across various operational domains can include addressing time critical system of systems (SoS), such as testing autonomous system adaptation platforms for understanding and assessing the impact of adding additional system functionalities [38], mission critical weapon SoS [39], assessing complexity in manufacturing systems [40], identifying software code complexities [41], and assessing commercial off the shelf products [42].