(1985). Levine's Atlas of Corporate Interlocks. Connections ... - INSNA

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M = 1 .0 - (g s /c s )/(g d/c d )
where :
g is the sum of all links
density in the matrix .
c is the sum of the cells contained in all sparse blocks .
s
gd is the sum of all links in blocks of density higher than the overall
density .
c d is the sum of all cells in dense blocks .
in blocks of density less than the over-all
If the density of all blocks is the same throughout the matrix then
the ratio of the mean density of sparse blocks to the mean density of the
dense blocks would be 1 .0 and the value of M would be 0 .0 indicating no blocking
. On the other hand, if the mean density of the sparse blocks is exactly
zero, then the value of M would be 1 .0 indicating optimal blocking .
As with CHB B, pairs of 500 CONCORR blockings of stacked 16*16 random
matrices with an achieved density of 0 .1750 and 0 .1083 respectively were
conducted and the proportional concentration of links in dense blocks (i .e .,
M) computed for each blocking . As with CHB B, the measure was computed both
for the random blocks and for the random distribution of links within these
blocks . Table 2 sets out the values of these two measures and that for the
set of data derived from the homeless families study as a contrast .
Table 2 . Proportional Concentration of Links in Dense Blocks
Number
Blocks
Random
Links
Random
Blocks
Real
Data
0 0 .000 0 .000 0 .000
4 0 .354 0 .488 0 .987
9 0 .518 0 .660 0 .993
16 0 .642 0 .767 0 .987
25 0 .723 0 .830 0 .989
36 0 .797 0 .885 0 .995
49 0 .875 0 .932 0 .995
64 0 .954 0 .984 O .a
a It needed only six partitions to achieve a solution of three or less
elements in each block in the real data .
Unlike CHB B the trend of the proportional concentration of links does
not increase in a linear fashion with increasing number of blocks over which
the measure is being computed . Instead it increases more rapidly in the
earlier stages of blocking and eases off as more and more partitions are
performed . The difference between the values for the random links as against
the random blocks decreases from 0 .134 to 0 .034 . The contrast with the