Kenneth Griggs and Rosemary Wild
influence over others); are there sub‐groups within the community that the network represents and, if so, what are they( e. g.,“ co‐worker” in Figure 1); which relationships or connections appear to be the most crucial to the functioning of the group and to the organization as a whole.
To understand how mathematical network analysis is used to answer such questions, a few definitions and concepts are in order.
A network graph is made up of nodes( also called vertices), and lines connecting the nodes( generally called edges). In Figure 1 each picture square represents a vertex and each line emanating from each vertex represents an edge. For example, each vertex might represent a government agency and the edges represent the connections or relationships each government agency has with other government agencies, whether local, state or federal.
In the simplest case, a network can be represented mathematically as an n x n symmetric matrix, called the adjacency matrix A, where n is the number of vertices in the network and the elements of the matrix A are the Aij such that:
A i j
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In the analysis of network data questions such as“ Who is the most important or influential person or group in the network?” are generally answered using centrality measures. The simplest and often most effective measure of the influence or importance of a node is the degree centrality, or simply degree, of the node. The degree of a node measures the number of edges( connections or relationships) associated with that node. In mathematical terms, the degree k i of a vertex i is:
=
Although the degree of a node may provide insightful information about the importance of a particular person or group within a network, it treats all connections as equal. It may be more effective to view the importance of nodes in a network not only by the number of connections associated with a node, but also the importance of the connections associated with a node. For example, a person within a government agency with relatively few direct connections to other employees may be shown to be“ important” by virtue of the fact that his / her connections are with other important and influential people in the business( e. g.,“ Joe” in Figure 1).
A centrality measure used to acknowledge that not all connections are equal is called an eigenvector centrality measure. In mathematical terms, if the centrality of vertex i is denoted by x i, then this effect can be measured by making x i proportional to the average of the centralities of i’ s network neighbors, where λ is a constant:
=
Thus the eigenvector centrality measure defines a centrality for each vertex that depends both on the number and the quality or importance of each of its connections or relationships. This is a critical factor in an SNA adoption model since it may be representative of the value of the whole social network to the organization.
Two other useful centrality measures that take network paths into account are closeness and betweenness centrality. Closeness is generally defined as the average geodesic distance to all reachable vertices, where a geodesic path is the shortest path between a specified pair of vertices. Betweenness is a crude measure of the control vertex i exerts over the flow of information( or other commodities) among other vertices( Newman, 2003). In a collaboration context, if the information flowing among participants takes the shortest path in the
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