1.1 Background
During the past two decays, the “network science” has attracted many attentions from mathematician, physicist, and sociologist. All the structures we mentioned below are structures existing in nature and human activity. Internet, a collection of computers linked by data connections. Food web, depicts feeding connections in an ecological community. Social network, people are connected if one knows another one. Lexical networks, the words are linked if they exist in a sentence. Neural network, neurons are connected by synapses. Basically, all these structures can be viewed as a combination of the individuals, and there are interactions between those individuals. “Network” is a science of a simplified representation to reduce a structure capturing only the topology properties. In this manner, we can identify their characteristics and simulate their behaviors in many different conditions.
The history of network can be traced back to Königsberg Bridge Problem which was solved by Leonhard Euler. There were seven bridges across the river which is through the city of Königsberg. The problem is “does any single path crosses seven bridges exactly once which is called Eulerian path exist?”. Figure 1 shows the map and the network structure of the problem.
Figure 1(a) is a map of 18th century Königsberg. Figure 1(b) is a simplified pattern of Königsberg. Figure 1(c) is the network structure of Königsberg Bridge Problem. This figure is from [1]
Figure 1 : Königsberg bridges problem and its network structure
In Figure 1(c), we can see the network structure of the Königsberg Bridge Problem.
The solution of this problem can be simplified as below: the Eulerian path traverses each edge once. These kinds of paths should enter and leave the nodes which are passing through except the source node and the end node. It means there can be at most two nodes having odd numbers of degree in network language. However, all the nodes in Figure 1 have odd degrees so that there is no Euler path existing in the structure.
The mathematical tools used to solve this problem is consider to be the first theorem in graph theory which is used to described the network structures by the researchers who study networks nowadays. These basic mathematical tools of network will be described reference in 2.1 and 2.2. These descriptions are mostly from [2]. In this manner, the well-known Erdős–Rényi model of random graph was developed by Paul Erdős and Alfréd Rényi. The properties of the random graph do not match the real data . We will discuss these differences in 2.3.
There are some modern features of the science of network indicated in [1]., they are :
Focus on the real world network and concern the theoretical and empirical questions
View networks as system evolving in time according dynamical rules
The example is like the “small world experiment”[3] in Sociology which is a empirical study. In 1998, Duncan J. Watts and Steven H. Strogatz published their famous paper, “Collective dynamics of small world networks”[4], and the model they presented is the first model realizing the properties of the small world networks via a simple dynamical rule. We will explain it clearly in 2.4.1. The other properties the researchers found in empirical data of networks is “scale-free degree distribution.”
In 1999, Reka Albert and Albert-Laszlo Barabasi gave the first model in their paper
“Emergence of Scaling in Random Networks” to reproduce the observed scale-free degree distribution in real data. The model is called BA model in the community of network science. The model is composed of a network structure and two generic mechanisms which we will discuss this more clearly in 2.4.2.
There is another research domain about the robustness of the complex structure. It was found that the scale-free networks have a higher degree of error tolerance than random network (Erdős–Rényi model), but error tolerance comes at a price in that the scale-free networks are extremely vulnerable to attacks (that is, the removal of some nodes that play important roles in the network connectivity. For this purpose, we can remove the nodes having highest degree or betweenness.)[5, 6]. There are many infrastructures composed of network structures, such as telecommunications, gas and water supply, transportation, and power grid. Power grid can be represented as a network of n nodes and k edges. Nodes are generators, substations and transformers and the edges are transmission lines. The power grid is vulnerable to natural disasters and physical attacks. It means the nodes and edges having a high probability to be remove from the power grid because they fail. We will review
previous research about the vulnerability and cascading failure in power grids and apply them on a assuming power grid from [7].
1.2 Purpose
In this thesis, we review the history of the complex network and test that does CTTC network have the properties we observed in many real network data. Further, we apply three methodologies test the robustness of a imaginary power grid.
The thesis can be divided into two parts: (1) analysis of CTTC network (2) vulnerability analysis of power grid. In (1), The purpose is to explain the structure of CTTC network and compare the real data with two models: BA model and EBA model. In (2), we apply the topology efficiency vulnerability and two cascading models in network science to see if we can identify the most vulnerable lines in an assuming power grid.