Matrices (in Social Network Analysis)

Generally speaking, a matrix is a rectangular arrangement of a set of elements or entries such as numbers or symbols that are arranged in rows and columns. The dimensions of the matrix in Figure 1 are three by four (3 × 4), as there are three rows and four columns. When discussing matrices, it is conventional to designate the number of rows as “m” and the number of columns as “n” and refer to the rows before the columns when describing the full size of a matrix. For example, Figure 1 displays a 3 (rows) × 4 (columns) matrix, which designates its full size. This entry describes the way matrices are utilized in social network analysis and defines common terminology such as ways and modes.

Figure 1 Example of a 3 x 4 matrix

In social network analysis, the most commonly used form of a matrix is the adjacency matrix. It is called an adjacency matrix because the entries [Page 1028]indicate whether two nodes are adjacent or not. Most social network matrices are square with as many rows and columns as there are nodes in a data set. The elements or entries in the cells of the matrix record information about the ties between each pair of nodes. An adjacency matrix may be symmetric or asymmetric. For example, the matrix in Figure 2 represents a friendship network. The rows represent the source of directed ties, and the columns the targets. Node 1 nominates Nodes 2 and 3 as friends, but Node 3 does not reciprocate the friendship nomination. Therefore, this is an asymmetric matrix with directed friendship ties. If the ties represented in the matrix were undirected (e.g., ties representing the relation “is married to” or “talked to” where direction does not make sense), the matrix would necessarily be symmetric. The simplest matrix is binary, which means that if a tie is present, the numeral 1 is entered in a cell, and if there is no tie, 0 is entered. The first row and first column are not really parts of the matrix in Figure 2, but social scientists typically show their data as an array of labeled rows and columns for presentation purposes.

Figure 2 Asymmetric adjacency matrix

Adjacency matrices of graphs are always square, as the one in Figure 2. They are also called one-mode matrices, as both the rows and columns refer to the same single set of nodes. However, in a two-mode matrix, the rows and columns refer to different sets of nodes. For example, imagine the nodes in the matrix rows of Figure 2 voting for different election candidates rather than selecting friends among one another. In this case, the columns would correspond to different candidates.

Matrices can be described as having ways and modes. The ways represent the dimensions of the matrix, such as when there are rows and columns, whereas the modes represent kinds of entities. A three-way matrix would then have rows, columns, and levels. Going back to the election example, suppose a researcher has data indicating which persons voted for particular candidates in different elections. This could be represented by a three-way, three-mode matrix, as in a data cube. Most studies, however, employ one-mode matrices that are the simplest to use. Overall, graphs of networks can be represented in matrix form, and mathematical calculations can then be performed to summarize the information in the graph that is useful in unpicking patterns of ties in social networks.

...