### 2.1 Network Theory

A network is a set of relationships. More formally, a network contains a set of objects (in mathematical terms, nodes) and a mapping or description of relations between the objects or nodes. The simplest network contains two objects, 1 and 2, and one relationship that links them. Nodes 1 and 2, for example, might be people, and the relationship that links them might be “are standing in the same room.”

There are also directional relationships such as 1 likes 2

In this simple network, the relationship could be symmetrical or non-directional: 1 and 2 like one another, or

their liking is mutual.

There need not be just one relationship mapped between nodes 1 and 2. For example, 1 and 2 might be in the same room and might like one another. When there is more than one relationship, this is called a multiplexrelationship.

Aside from their directionality, or lack of it, relationships might be more than the sharing of an attribute or being in the same place at the same time. There can be a flow between the objects or the nodes. Liking, for example, might lead to an exchange of gifts. Flows and exchanges can be very important in network theory.

At one level, this list of concepts of relationships between pairs of nodes is now logically complete. But consider a network between pairs that operates via an intermediary node. For example:

1 is connected to 3 via 2. The relationships shown are directional and not reciprocal, but they need not be. They could be non-directional or reciprocal. Consider a non-directional or reciprocal three node relationship in which 1 and 2 like one another, and 2 and 3 like one another. The network connection may be represented by a positive sign:

One can describe the network distance between pairs of nodes in terms of the number of steps or links between them. There are obviously two steps between 1 and 3. But if 1 also likes 3, as shown below, the network is said to be “transitive” or balanced (see below), and in this case all three nodes are directly linked.

### 2.1.1 Information in network theory

While the availability of suitable relational data was for a long time the bottleneck that limited the expansion of the relational approach, a completely new situation has arisen with the progressive digitalisation of society through the Internet.Relational information, which was often only available in printed form, is now available online: Databases and in particular the data generated by the new communication technologies of the Internet make it possible to examine social processes and systems in previously unknown details.

With the growth of the analytical potential of network analysis, graphic forms of network visualization have spread almost automatically, without in-depth discussion, and today they are an integral part of many scientific publications. Small networks have always been analyzed by pencil or crayon. Only the automatic arrangement of the units into a layout, which allows a variety of properties of the underlying graph to be readable, allows the significance of the arrangements to be inspected more closely by mapping further features and attributes.Network analyses can analyze the individual action contexts of the actors, certain subsystems of the network or the entire system. In the case of large networks at the latest, it is not easy to survey these multiple descriptions in numerical form. Only when different calculations are presented simultaneously do the peculiarities of the embedding of the actors become clear.

The new graph-theoretical methods have their roots in methods of traditional multivariate analysis (MDS). They make it possible to analyze the Network analysis. A growing paradigm 219 spatial arrangements of mathematical spaces flexibly non-metrically, while preserving the neighborhoods of the metric solutions. For the human eye, these simplifications do not pose any problem worth mentioning: For humans it is easy to take relevant information about the neighborhoods from the representations.Graphical methods are particularly suitable for communicating diverse information, since different information can be communicated simultaneously on different graphic channels: the position of the units, their sizes and colors. Quality features can be represented by shapes and quantitative attributes using appropriate color schemes.

2.1.2 Illustrations of network theory

Relational individual observations of the most diverse processes can be arranged with network technologies in order to generate representations of the social systems formed by them.Under positive conditions, even fundamental changes in companies can be analyzed, which, like the capital linkages among the one hundred largest German companies, can hardly be surveyed in numerical form. The following presentations are based on the main opinions of the German Monopolies Commission and provide insight into profound changes in the German economic system.

Table 3: Capital links of the one hundred largest German companies in year 2000