3.1 From Graph to Matrix

While Chapter 1 was dedicated to exploring the various ways real world social networks can be represented in graph form, this chapter will focus on using a matrix as a second way of representing networks.

Networks represented as graphs feel quite intuitive to us. We feel as if we can get a good sense of the network by looking at simple graphs. Networks as a matrix however are a bit more abstract. They are far more mathematically useful however. With many tens, hundreds, thousands, or millions of nodes and edges, creating a graph with these amounts of data only results in what network analysts call a “hairball.” Nothing can be understood through intuition. Thus, it provides us more analytical leverage when we switch to representing the network as a matrix. When we represent the network as a matrix, we are able to efficiently calculate features of the network that we would not be able to see with our eyes alone.

A matrix is, quite simply, a set of attributes that represent the values of a particular case. Breaking that explanation down, we can imagine a matrix as follows in Table 0.1

Table 0.1: Example of a general matrix.

Atribute 1 Attribute 2 Attribute 3
Case 1 Value 1 Value 4 Value 7
Case 2 Value 2 Value 5 Value 8
Case 3 Value 3 Value 6 Value 9

This common matrix presented in Table 0.1 seems similar to a spreadsheet. Well, that is because a spreadsheet can be thought of as a matrix! If you have ever seen statistical regression models, this is actually what a computer is doing in converting the values in the various cells in to a regression equation! It is all quite cool, but not exactly what we are going to need going forward in studying the basics of network analysis.

Instead, we will be focusing on a particular type of matrix called a relationship matrix. A relationship matrix is when, instead of asking what value of an attribute a case has, we ask about the value of describing how a case relates to another case. Thus, we create a matrix with the following organizational properties as shown in Table 0.2.

Table 0.2: Example of a relationship matrix.

Case 1 Case 2 Case 3
Case 1 Value 1 Value 4 Value 7
Case 2 Value 2 Value 5 Value 8
Case 3 Value 3 Value 6 Value 9

A relationship matrix thus captures exactly that, the relationship between two cases. In Figure 2.2, the relationship between Case 1 and Case 2 is Value 4, and the relationship between Case 2 and Case 1 is Value 2. Wait, would that mean Value 2 and 4 are the same?

The answer is maybe. Depends on what type of relationship is being captured. In the case of a matrix, we generically call the rows i while calling the columns j (this is denoted in Table 0.2).. By convention, we say that i sends to j, so if the relationship matrix was capturing friendship, we might say that i considers j to be a friend (sends the consideration) and so if i is Case 1 and j is Case 2, that would be Value 4. But when i is now Case 2 and j is Case 1, we are now asking if Case 2 considers Case 1 to be a friend. That could be true, or it could not be.

Furthermore, we describe the size of the matrix by the number of rows by the number of columns. n is used to represent the number of rows and m is the number of columns. Thus, the above table is described as a 3x3 graph. The question of a graphs size, nxm, is of great importance in conducting certain mathematical operations, as will be discussed below.

There are many types of relationship matrices as the basic principles just stated can be varied to capture different underlying facets of relationships. This chapter will cover the adjacency matrix, and the reachability matrix as examples of how different types of networks can be best captured in a matrix form and then manipulated to produce sociologically meaningful results. Another type of matrix, the two-mode matrix, will be covered in a subsequent chapter.