Network Matrices

Network matrices are defined by Robert A. Hanneman and Mark Riddle (2005) as collections of elements into rows and columns, and they are often used in network analysis to represent the adjacency of each actor to each other actor in a network. To talk about network matrices, it requires some background information on networks and graph theory. According to Stephen P. Borgatti and colleagues (2018), “networks are a way of thinking about social systems that focus our attention on the relationships among the entities that make up the system, which we call actors or nodes” (p. 2). In the case of social sciences, actors or nodes are typically people. These actors have characteristics, such as gender and race/ethnicity, and the relationships between them also have characteristics, for example, friendship ties and support ties. Borgatti and colleagues (2018) argued that part of the power of the concept of a social network is that it provides a mechanism, which they call “indirect connection” (p. 2) by which disparate parts of a system may affect each other. In extent, social network analysis is a social science that is primarily concerned with the structure of social networks and the position of actors within those networks.

Networks can be represented mathematically as graphs that consist of two sets: a set of vertices (nodes or actors) and a set of edges (links or ties) connecting the vertices.The field of mathematics concerning the study of graphs is called “graph theory.” In social network analysis, networks may be represented as graphs or matrices. 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 Matrix 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, Matrix 1 is a 3 (rows) × 4 (columns) matrix, which designates its full size.

In social network analysis, the most commonly used form of matrix is the adjacency matrix. It is called “adjacency matrix” because the entries 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 Matrix 2 represents a friendship network. The rows represent the source of directed ties, and the columns [Page 1042]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 are 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, a one (1) is entered in a cell and if there is no tie, a zero (0) is entered. The first row and first column are not really parts of the matrix in Matrix 2, but social scientists typically show their data as an array of labeled rows and columns for presentation purposes.

...