# 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
#### 5.3.1 Preferential Attachment Networks  
Generated networks using the NetLogo software (version 5.3.1) illustrate the behavior of real-world networks (such as connection from and to a website, social networks, and collaboration networks), where few nodes have a large number of connections, whereas all the other nodes have only a few. This phenomenon in which network nodes prefer to connect to the popular existing nodes is called preferential attachment. This model starts with two nodes connected initially, and thereby every new originating node randomly picks an existing node with which to connect, with some inherent bias, i.e., the chance of a node being chosen to be connected to by another node is directly proportional to the number of connections (degree) it already has. The networks that arise from this phenomenon often follow a power law distribution, i.e., the distribution of the number of connections of each node is not normal [34]. [PERSON] and [PERSON] originally coined this mechanism for creating scale-free networks, so the networks created by this mechanism are called Barabasi scale-free networks. Figure 3 illustrates the evolution of interactions in the network over fifteen different snapshots. To analyze the flow of information generated from one node to another of the network based on the individual node degree evolution over several time-steps of network formation, the concept of transfer entropy (TE) is used. To generate TE values based on individual node degree evolution, data on the number of nodes present along with the individual node degrees over 15 different time-steps were captured using NetLogo [35]. Table 2 illustrates the degree evolution of nodes portrayed in Figure 3.  
To calculate the TE values (see Table 3), joint probabilities are calculated for the emerging node degrees of Table 2. Thus, to calculate the TE from node 0 to node 1 at time-step t1, we need to find the probability \(p\) (y2, y1, x1) as required for calculating TE, where y2 corresponds to the value located in the second column, second row; y1 corresponds to the  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c} \hline \hline
**Time-Steps** & **0** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** & **12** & **13** & **14** \\
**Node** & **0** & **1** & **2** & **1**  
value in the first column, second row; and x1 corresponds to the value in the first column, first row; i.e., \(p\) (1,1,1). We now count the number of matching combinations of these values that exist in the two rows corresponding to node 0 and node 1. In the graph (Figure 4), the X-axis represents the node numbers and the Y-axis represents the corresponding TE values. It is read as: for the red line, Mark 0 is the TE value from node 0 to node 1, Mark 1 is the TE value from node 1 to node 2, and so on. For the blue line, mark 0 is the TE value from node 1 to node 0, mark 1 is the TE value from node 2 to node 1, and so on. It is seen that, in general (see Figure 4), information flow from the source nodes to the destination nodes initially rises and then dominates information flow from destination nodes to source nodes. At the early stages of the graph, the trend line of S-D, when observed, shows that TE between node 0 and 1 is zero with node 0 as the source and node 1 as the destination. This simply suggests to the initial network structural formation where at the initial time-step nodes 0 and 1 are connected, on which the simulated network builds over the considered time series. Before calculating the network entropy measure, we introduce a measure of connectedness, also called the reproductive number in complex networks. The reproductive number, R\({}_{0}\), helps to understand the context of information spread in a network [36]. R\({}_{0}\), also known as the connectedness of a network, can be calculated as follows. Consider a node \(n\) with degree \(k\) to communicate an idea or spread information to its neighbors with a probability \(r\).