degree of centralization

Examples of degree of centralization in the following topics:

• Hypotheses about the means of two groups

  • Using the Knoke information exchange network, we've run Network>Centrality>Degree, and saved the results in the output file "FreemanDegree" as a UCINET dataset.
  • Let's perform a simple two-sample t-test to determine if the mean degree centrality of government organizations is lower than the mean degree centrality of non-government organizations.
  • The normed Freeman degree centrality measure happens to be located in the second column of its file; there is only one vector (column) in the file that we created to code government/non-government organizations.
  • We see that the average normed degree centrality of government organizations (75) is 6.481 units higher than the average normed degree centrality of non-governmental organizations (68.519).
  • Test for difference in mean normed degree centrality of Knoke government and non-government organizations

• Study questions

  • Why is an actor who has higher degree a more "central" actor?
  • How does Bonacich's influence measure extend the idea of degree centrality?
  • What kinds of approach did each use: degree, closeness, or betweenness?
  • Which actors have highest degree?
  • Can you think of a real-world example of an actor who might be powerful but not central?

• Freeman's approach

  • Linton Freeman (one of the authors of UCINET) developed basic measures of the centrality of actors based on their degree, and the overall centralization of graphs.
  • Figure 10.5 shows the output of Network>Centrality>Degree applied to out-degrees and to the in-degrees of the Knoke information network.
  • That is, what does the distribution of the actor's degree centrality scores look like?
  • In the current case, the out-degree graph centralization is 51% and the in-degree graph centralization 38% of these theoretical maximums.
  • Freeman degree centrality and graph centralization of Knoke information network

• Bonacich's approach

  • Phillip Bonacich proposed a modification of the degree centrality approach that has been widely accepted as superior to the original measure.
  • One begins by giving each actor an estimated centrality equal to their own degree, plus a weighted function of the degrees of the actors to whom they were connected.
  • First, we examine the case where the score for each actor is a positive function of their own degree, and the degrees of the others to whom they are connected.
  • This is a straight-forward extension of the degree centrality idea.
  • The Bonacich approach to degree based centrality and degree based power are fairly natural extensions of the idea of degree centrality based on adjacencies.

• Introduction to degree centrality

  • So, a very simple, but often very effective measure of an actor's centrality and power potential is their degree.
  • With directed data, however, it can be important to distinguish centrality based on in-degree from centrality based on out-degree.
  • Actors who display high out-degree centrality are often said to be influential actors.
  • Simply counting the number of in-ties and out-ties of the nodes suggests that certain actors are more "central" here (e.g. 2, 5, 7).
  • We can see "centrality" as an attribute of individual actors as a consequence of their position; we can also see how "centralized" the graph as a whole is -- how unequal is the distribution of centrality.

• Characteristics of Organizational Structures

  • Important characteristics of an organization's structure include span of control, departmentalization, centralization, and decentralization.
  • Each of these structures provides different degrees of four common organizational elements: span of control, departmentalization, centralization, and decentralization.
  • Centralization is usually helpful when an organization is in crisis and/or faces the risk of failure.
  • Centralization allows for rapid, department-wide decision-making; there is also less duplication of work because fewer employees perform the same task.
  • This diagram compares visual representations of a centralized vs. decentralized organizational structure.

• Flow centrality

  • The flow approach to centrality expands the notion of betweenness centrality.
  • By this more complete measure of betweenness centrality, actors #2 and #5 are clearly the most important mediators.
  • While the overall picture does not change a great deal, the elaborated definition of betweenness does give us a somewhat different impression of who is most central in this network.
  • Some actors are clearly more central than others, and the relative variability in flow betweenness of the actors is fairly great (the standard deviation of normed flow betweenness is 8.2 relative to a mean of 9.2, giving a coefficient of relative variation).
  • Despite this relatively high amount of variation, the degree of inequality, or concentration in the distribution of flow betweenness centralities among the actors is fairly low -- relative to that of a pure star network (the network centralization index is 25.6%).

• Introduction to closeness centrality

  • Degree centrality measures might be criticized because they only take into account the immediate ties that an actor has, or the ties of the actor's neighbors, rather than indirect ties to all others.
  • One actor might be tied to a large number of others, but those others might be rather disconnected from the network as a whole.
  • In a case like this, the actor could be quite central, but only in a local neighborhood.
  • Closeness centrality approaches emphasize the distance of an actor to all others in the network by focusing on the distance from each actor to all others.
  • Depending on how one wants to think of what it means to be "close" to others, a number of slightly different measures can be defined.

• Effects of Linear Transformations

  • This section covers the effects of linear transformations on measures of central tendency and variability.
  • Recall that to transform the degrees Fahrenheit to degrees Centigrade, we use the formula
  • The formula for the standard deviation is just as simple: the standard deviation in degrees Centigrade is equal to the standard deviation in degrees Fahrenheit times 0.556.
  • Since the variance is the standard deviation squared, the variance in degrees Centigrade is equal to 0.5562 times the variance in degrees Fahrenheit.
  • will have a mean of bμ+A, a standard deviation of bσ, and a variance of b2σ2.

• Radians

  • Radians are another way of measuring angles, and the measure of an angle can be converted between degrees and radians.
  • One radian is the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle.
  • Since we now know that the full range of a circle can be represented by either 360 degrees or $2\pi$ radians, we can conclude the following:
  • As stated, one radian is equal to $\displaystyle{ \frac{180^{\circ}}{\pi} }$ degrees, or just under 57.3 degrees ($57.3^{\circ}$).
  • Explain the definition of radians in terms of arc length of a unit circle and use this to convert between degrees and radians