Structural Analysis With Graphs

Graphs

Graph is a mathematical term, which provides us a representation method that is completely equivalent to sociomatrices and adjacency matrices mentioned in the previous chapter. In its most basic and theoretical sense, using one or other of these methods does not contribute to how easily we gather information on or how well we represent a social network phenomenon. Matrix algebra has been a stronghold of mathematics and was the choice of early social network researchers. However, analysis methods using the graph theory have started to develop substantially along with the advancement of mathematics and computer science in recent decades. Therefore, learning how to use graphs to represent social networks is important not for theoretical aspects but for practical purposes, i.e. using these methods in practice. While dealing with the question of how structural features are found from the data set, we will also see how they are reflected in graph and matrix features simultaneously. As a result, your early exposure to different social network representation methods with their corresponding analysis tools would make it easier for you to develop an oversight and benefit from books and articles written within the scope of different disciplines and in different time periods.

The mathematical concept of a graph is a structure which consists of “vertices” which are connected by “edges”. As the mathematical term is defined regardless of what it is meant to represent, mathematicians named the components in the graph as geometric terms. They further differentiate between edges and “arcs”, a term used for the ties in a directed or asymmetrical relation (Valiente 2002). Software people prefer the word “node” instead of vertex, and “tie” instead of edge or arc. Social researchers would feel home with the term “actors” instead of a mathematical or computational term like “vertices” or “nodes”, and “relation” or “tie” instead of edge or arc. Figure 1 shows a visual of a fictional personal friendship network. As far as our example is concerned people are vertices and the friendships among them are edges of the graph in this visualization.

Here, I will opt for the terms “actor” and “tie” to maintain a style in accordance with the social sciences perspective, but I will also use the more neutral term “component” instead of “actor” from time to time while handling theories on graphs. The reason for this is that not every socio-economic component is “active” or is an “actor” and that some of them might be passive as well, e.g. the “country” in the case of trade network between countries. Even though the varying terminology does not affect the underlying mathematical processes, it allows us to talk about phenomena in various and appropriate ways. It takes some terminological versatility to use the same mathematical apparatus when talking about, for example, electrical networks of a city versus a network of political relations.

The mathematical framework used for the representation of graphs is also quite suitable for social networks. Furthermore, it is more clear and more compact in representing social networks, when compared with sociomatrices. Let us see this method at work in an example. The example below is a social network data file written in the particular graph format of the Pajek software, which is commonly used as input format in many other software packages as well:

*Vertices 4
1 “Albert”
2 “Richard”
3 “Betty”
4 “Louis”
*Edges 
1 3 7
1 4 11
2 1 12
2 3 21

Even without a detailed explanation of the data format used, the data above can be read easily by looking at the English terms regarding graphs, if you are familiar with graph terminology. The data format shown here is used in the Pajek software (Nooy, Mrvar, and Batagelj 2005) and is compatible with many social network analysis software developed later on. You can save the graph representation above as a file named “example.net” and then open the file with Gephi for visualization and analysis purposes. Similarly, you can also export one of the sample social networks included in Gephi (and visible on the opening screen) in this format and then use it. Relatively newer software such as Gephi recognize the formats of Pajek and other preceding programs, define their own usually-richer data formats and allow the conversion between two formats.