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

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where di is the density in dense block i
dj is the density in sparse block j
dt is the over-all density
k is the number of dense blocks
1 is the number of sparse blocks
ci is the number of cells in dense block i
cj is the number of cells in sparse block j
c t is the total number of cells in the matrix
In effect CHB B is the sum of the squared differences between dense
block densities and over-all density in a matrix weighted by the proportion of
cells in the dense blocks and by the reciprocal of the squared complement
of the over-all density added to the squared differences between the densities
in sparse blocks and the over-all density weighted by the proportion of
cells in the sparse cells and the reciprocal of the squared over-all density .
It is easy to show algebraically that if there is no blocking at all,
then the individual block densities are exactly the same as the over-all densities
and CHB B = 4 .0 . It follows that the CHB B is a measure of 'fit' in
the sense that it only achieves the value of 1 .0 when the dense blocks are
all precisely 1 .0 and the sparse blocks exactly 0 .0 . It is, in fact, a
measure of the extent to which the blocking approximates the 'fat' fit condition
.
MonteCarlorunsonCHBB
It is of interest to know how CHB B behaves when there is in fact no
intrinsic blocking in the data . In order to test this CHB B was computed
for blockings of 500 pairs of purely random zero-one matrices . A program
was written which allocated links at random in rows and columns of a matrix
(or matrices) of a specified order to a specified density of links (either
symmetric or asymmetric) . In the data reported here two asymmetric matrices
of order 16*16 were allocated random links to a density of 0' .1750 and
0 .1083 respectively . These densities were chosen so as to duplicate those
in an empirical set of data against which I wished to compare the results .
Each matrix was created using a random procedure to allocate entries to offdiagonal
cells of the matrices . The resulting matrices were then partitioned
using CONCOR . Joint CHB B was calculated for each partition and
each partition repartitioned until the resulting partitions were made up of
3 elements or less . Since CONCOR will find blocks even if the links are
purely at random, the CHB B were recalculated at each partition by accepting
the size of the blocks - i .e ., as determined by the partition of the
two random matrices but by allocating elements to each partition purely
at random insofar as the calculation of densities were concerned . Therefore
although block sizes remained constant, it was purely a random process
whether links were allocated to particular blocks or not in order to calculate
densities . I refer to random blocks as those derived from the
blocking of the original random matrices aiid random links as links ascribed
to random blocks purely at random to counteract the effect on calculations
to the unavoidable modifications of density due to CONCOR procedures .
Table 1 sets out the mean value of CHB B at each partition of the matrix .