Properties of Social Networks

There are also properties of the network as a whole. I will shortly present those concepts which are considered in this thesis and then discuss how they are related to each other and what facts might be important for my analysis.

2.4.1. Centralization and Density

Once the centrality of the nodes within a network has been determined, one can use this information for calculating the overall centralization of a network. It is based on the differences between the centrality of the most central node and that of all other nodes.

It tells us if the network is organized around particular focal points (Scott 2000:90).

The density of a network is the number of ties present, divided by the number of ties possible. It describes the overall level of linkage or cohesion among the nodes of a network (Scott 2000:69-71).

In non-valued networks, density and centralization can vary from 0 to 1 (or 0% to 100%). But they are not independent of each other. Limitations are imposed by definition as well as by natural boundaries.

The most central network one can think of contains a node that has direct connections to all other nodes, while those other nodes do not have connections which each other (see figure 2-3). This, by definition, makes it impossible to have a network with an extremely high density and centralization at the same time.

Centralization is based on the fact that links are missing and that they are missing unequally.

The lowest centralization one can think of is a network in which all nodes have the same number of connections to others. For example, this is the case when each node is connected to only two other nodes. One can draw this as a circle or wheel (see figure 2-4) in a graph.

Obviously, this imposes a minimal density on a

network of a given size. As soon as only one link is missing, the number of connections per node becomes unequal and the centralization is not zero anymore.

Figure 2-3: a highly centralized graph

Figure 2-4: a highly

decentralized graph Figure 2-5: another decentralized graph

A network in which everybody is connected to everybody has a centralization of zero and a density of one (see figure 2-5). However, this extreme case is not realistic either, at least not in large social networks. To give an example, it is improbable that 80 million people living in Germany know each other. Obviously, there is a natural limit on how many social contacts a person can have. Therefore, you will not find a large social network with a density near one, although it is at least possible by definition to have such a network.

2.4.2. Scale-free Networks – Lessons from Other Sciences

In the past, it was quite common to work with randomly generated graphs as models for complex networks. A whole paradigm developed around the work of Erdős/Rényi (1960). However, as scientists have shown recently, many networks existing in reality are not random at all (see Barabási/Bonabeau 2003 for an overview).

Many networks, man-made as well as in nature, tend to be “scale-free”. This means that they are dominated by a few highly connected nodes. Barabási/Bonabeau (2003:53) use the U.S. airline system as an example (see figure 2-6 on the right). Airports like Chicago, New York or Atlanta serve as “hubs” that have a very high number of connections, while most other airports are much less important with only a few connections each. The number of links of the nodes in such a network approximates a power law distribution.

Figure 2-6: random and scale-free networks compared – graph from Barabási/Bonabeau (2003:53) graph removed for the publication of this document because of copyright reasons

You can download the article containing this graph for free on the author's website at:

http://www.nd.edu/~networks/Publication%20Categories/publications.htm

Nicolas Marschall

Scale-free networks can be found in many places, from the protein regulatory networks in cells to the Internet (Barabási/Bonabeau 2003). When a network is scale-free, it implies a high centralization, as centralization is the difference in degrees between the actor with most degrees and all others. In contrast, nodes in random networks all tend to have about the same number of links, while “it is extremely rare to find nodes that have significantly more or fewer links than the average” (Barabási/Bonabeau 2003:52). The number of links of these nodes approximates a bell-shaped Poisson distribution. Such networks are also called “exponential” because the probability of finding a node with an exceptionally large number of links decreases exponentially the higher this number gets.

Exponential networks exist, for example, in road networks where at crossings more than four directions are unusual (see figure 2-6 on the left).

The main argument in the literature is that scale-free networks are significantly different from exponential networks in their “error and attack tolerance” (Albert et al 2000). This terminology originates from physics and computer sciences where attacks on computer or electricity networks are being modeled (e.g. in Holme et al 2002). On the one hand, scale-free networks are quite robust when nodes are removed at random, but on the other hand they are highly vulnerable when attacked intentionally (Callaway et al 2000:5471).

For example, from time to time airports have to be closed temporarily because of bad weather conditions or accidents. Let us assume this happens at random. Then most likely it will hit one of the many unimportant local or regional airports, and the airline network as a whole will not be interrupted. However, if terrorists wanted to interrupt the air transportation system of a country, they would most certainly choose one or more of the hubs as target for their attack. If Chicago, Atlanta and New York airports are being closed down because of bomb threats, the air transportation network of the United States would indeed be interrupted.

In random networks where all nodes have approximately the same importance, there is not such a difference between random errors and intentional attacks.

This research has much in common with my approach. While a social scientist might ask the question: “What are the consequences for my policy network if the most important actor in environmental politics does not respond to my questionnaire?”, a computer scientist may ask: “What happens if the most important hub on the Internet is

being shut down by hackers?”. Both cases can be modeled with network analysis and the methodological approaches towards both problems are surprisingly similar. In both cases, the consequences of the removal of a node from the network are investigated.

To conclude, the research in physics (e.g. Albert et al 2000) shows that the underlying distribution of node degrees is a relevant influencing factor for the vulnerability of a network towards the removal of nodes.