Chapter 4 Lines, Direction and Density

In the previous chapter I looked at the ways in which relational data can be handled and managed in matrix form. Many fundamental features of social networks can be analysed through the direct manipulation of matrices - the transposing, adding and multiplying of matrices all yield information on their structure. Matrix algebra, however, is rather complex for most researchers (but see Meek and Bradley, 1986). Although matrices are useful for the organization and storage of relational data, specialist computer programs allow an easier and more direct approach to network analysis. The available packages implement a variety of analytical procedures, and any user of the programs must have some understanding of how they work.

A common framework for social network analysis programs is the mathematical approach of graph theory, which provides a formal language for describing networks and their features. Graph theory offers a translation of matrix data into formal concepts and theorems which can be directly related to the substantive features of social networks. If the sociogram is one way of representing relational matrix data, the language of graph theory is another, and more general, way of doing this. While it is not the only mathematical theory which has been used for modelling social networks, it is a starting point for many of the most fundamental ideas of social network analysis.

It is the concepts of graph theory which figure as the principal procedures in the UCINET and GRADAP programs, though the readily accessible computer programs endeavour to keep as much of the mathematics as possible hidden from the user. Data in matrix form can be read by the programs, and suitable graph theoretical concepts can be explored without the researcher needing to know anything at all about the mechanics of the theory or of matrix algebra. Nevertheless, an understanding of graph theory will significantly help to improve the sophistication of a researcher's analyses, by ensuring that he or she chooses appropriate procedures. Indeed, GRADAP's data structure and management procedures require an understanding of basic graph theoretical ideas.

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Graph theory concerns sets of elements and the relations among these, the elements being termed points and the relations lines.' Thus, a matrix describing the relations among a group of people can be converted into a graph of points connected by lines. A sociograin, therefore, is a 'graph'. So far, this should be very familiar from what has already been discussed in Chapters 2 and 3. It is important to be clear about the difference between this idea of a 'graph' and the graphs of variables used in statistics and other branches of quantitative mathematics. These more familiar graphs we might term them 'graphs of variables' - plot, for example, frequency data on axes which represent the variables. The graphs of graph theory - 'graphs of networks' - express the qualitative patterns of connection among points. Indeed, graph diagrams themselves are of secondary importance in graph theory. As has already been suggested, it is often very difficult to draw a clear and comprehensible diagram for large sets of points with complex patterns of connection. By expressing the properties of the graph in a more abstract mathematical form, it is possible to dispense with the need to draw a sociogram and so make it easier to manipulate very large graphs.

Nevertheless, the drawing of graph diagrams has always been of great illustrative importance in graph theory, and many others will be used in this book. Because of the visual simplicity of small sociograms, I will begin with an introduction to the principles involved in drawing graph diagrams before going on to introduce the basic concepts of graph theory.