To get at the ideas of centre in networks, we'll be looking at both centrality of vertices and centralization of a network. Periphery is often a derived implication of centralization. In order to fix ideas for further discussion it is useful to keep in mind two ideal-typical and intuitive networks: star and line network. A star network is often taken to be the most efficient structure to connect vertices with a given number of links. Centralization, to be more precisely defined later, is always relative to star network.
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Centrality is a characteristic of vertices; we use centralization as a characteristic of network. The most obvious measure of centrality of vertices is degree centrality, which is defined as the immediate neighbours of a vertex. In the above example v1 has degree 5 and the rest of the vertices connected to it has degree 1. In the line network, vertices v7 and v12 each is of degree 1 and the rest is of degree 2.
To motivate network centralization, think of the star as an ideal centralized network and the line network as a less than ideal one. In a star network, the degree centrality of the vertex in the middle is very different to the degree centrality of the rest. The variation is less marked in a line network. This leads us to a definition of centralization as the variance of centralities which should be higher for a star network than for a line network of comparable size. And so, we derive measure of network centralization based on vertex centralities, in this instance degree centrality. To find degree centralities in Pajek, do as shown above.
This procedure applied to hawthorne-friend.net data gives Network Centralization shown in the Report window to be 0.19231. As mentioned earlier, centralities are characteristics of vertices and so they are given in a partition called All Degree partition in the partition area. The partition name also noted that it has 14 vertices and is derived from network 1 that is C:\temp\hawthorne-friend.net. Click on Edit Partition icon just to the left of the name of the partition (1. All Degree partition of N1 (14)) to see the degree (centrality) of each vertex.
You may have noticed that there are three choices to degree partition listed on Pajek menu: , and . They refer to indegree, outdegree for directed network and degree for (simple) network. They are of course different only in the case of directed network.
Two other centrality measures are based on a global or wider view of the network, as opposed to the local view of immediate neighbour. We need to use shortest path and its length (i.e. geodesic and distance, respectively) in our subsequent definition. Closeness centrality of vertex v is a summary measure of the distances from v to all other vertices; the number of other vertices divided by the sum of all distances between v and all others. Intuitively, shorter distances to other vertices should be reflected in a vertex's lower closeness score. In this sense, one can think of closeness as reflecting compactness. Note that closeness is not defined for network with separate components or network with isolates, i.e. a subset of vertices not linked to other subset of vertices.
The last measure of vertex centrality builds on the notion that a vertex is central if it is needed to connect other pair of vertices. This is called betweenness centrality. The word says it all; if not, brokerage notion may come to mind. Betweenness centrality of a vertex v is the proportion of all geodesics between the pairs of vertices which include v. The more a vertex is needed for, say, passing of information between a the pairs, the higher is its score. In this sense, one can think of betweenness as reflecting facilitation of circulation.
In Pajek, different centralities are not centralised in one menu. Because degree is an integer, it defines a partition in Pajek terms, hence it is in . On the other hand, closeness and betweenness are proportion or real numbers, so it is in .
If you are interested in geodesics and distance, look at and .
Methods for delineating the core of networks continue to be developed and often presented in the Social Networks and other journals. In fact, more developments, applications and resources are available for your own exploration beyond this academic field. You can start by following links collected here.
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© G Tampubolon - 17 December 2004