(1990). Ties and Bonds. Connections, 13 (3) - INSNA

aggregated distributions of centrality measures . Cumulative probability distributions are shown in Table 1 (n = 6)
and in Table 2 (n = 7) . The graphical representations of the probability distributions are shown in Figures 1 to 6 .
The sparseness of values made it difficult to determine precise significance levels for small size networks . At
the 5 % one-tailed, for example, the critical values for the sizes 6 and 7 are shown in Table 3 :
Table 3 shows that the closeness-based graph centrality has the largest critical value at 5% of significance
level . On the average, betweenness-based graph centrality has the lowest mean value, degree-based second, and
closeness-based graph centrality has the highest mean value . (For example, the means are 0 .34 for CD, 0 .27 for CB,
and 0 .40 for Cc, for n = 7 .)
2 . Effects of Number of Arcs
Recall that these measures of centrality are standardized so as to vary between 0 and 1, where the value 1
indicates the perfectly centralized network based on the asterisk-shaped model . Therefore, the perfectly centralized
network would be obtained only when the number of relations equals n-1 . As the number of relations increases, the
degree of centralization of a network decreases . Thus, there is an upper-bound of a graph centrality value for a
given number of relations for a given number of points . (See Table 4)
It is observed in Table 4 that the closeness-based centrality measure has higher upper limits than the other
two types of measures for a given number of relations and a given number of nodes .
In the comparison among different measures of centrality, the important parameter to consider is the number
of relations (or arcs), since each measure seems to be affected by it differently . I computed the mean of the
enumerated distribution for a given number of relations and for a given size of the network . Figures 7 and 8 show
the distribution of means of graphs of size 6 and 7 respectively .
Observation of these figures prompts the following statements : is as follows: when the number of relations is
small (close to n-i), the betweenness-based measure has the largest value on the average, the closeness-based
measure the next, and the degree- based measure the smallest . However, the values of CB seems to be more
susceptible to the change (increase) of the number of relations than the other two measures are . As the number of
relations increases, the order changes from CB > Cc > CD to CC > CD > CB, and this order remains until the graph
becomes saturated or becomes a clique. On the average, the closeness-based measure seems to be always greater
than the degree-based measure .
3 . Correlations
One way to compare measures of centrality is to apply correlation coefficients . For all possible connected
graphs enumerated or simulated, the correlation coefficients between a pair of measures of centrality were computed.
The result is in Table 5 :
These correlations show that the betweenness-based centrality measure behaves somewhat differently from
the other two measures, indicated by lower correlations . The degree-based measure and the closeness-based
measure are related very closely in a linear manner .
4. Comparison by Order of Magnitude
13
The above analysis is based on aggregated information, taking the means of distributions . I have also examined
each possible graph in terms of the order of magnitude of the three measures . The result is summarized in
Table 6.
Table 6 shows the proportion of graphs over all possible graphs which resulted in specified orders of
centrality measures. Overall, as we would expect from the previous aggregated observations, the large proportion of
graphs produce the largest Cc, CD second, and the smallest CB . Another noticeable order is CB > CC > CD .
Let me exemplify some graphs of the above two orders, Cc > CD > CB and CB > CD > Cc, which are shown in
Figures 9 and 10 .
For each graph ratios of different centrality values were computed and the above two examples produced
largest ratio values (meaning most extreme exemplars) . The observations of the above examples as well as many
other exemplars may be summarized as follows : The graphs whose order of centrality values is CB > CC > CD tend
to be divided into sub-clusters that are connected to each other by a line or via a focal point . This type of graph
may be characterized as decentralized network, in the manner defined in organizational research . On the other
hand, the graphs which produces Cc > CD > CB would be described as one- cluster network which contains a circle
involving a large proportion of points in the network . The exception of the above characterization of network
Fall/Winter, 1990
Connections