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holding B, C, and D fixed, and so on:
A ( X ( m×npq ) Z )
A
( AZ T Z ) + ,Z =( D
C
B )
B ( X ( n×pqm ) Z )
B
( BZ T Z ) + ,Z =( A
D
C )
C ( X ( p×qmn ) Z )
C
( CZ T Z ) + ,Z =( B
A
D )
D ( X ( q×mnp ) Z )
D
( CZ T Z ) + ,Z =( C
B
A ) .
Here is a small number like 10 9 that adds stability to the calculation
and guards against introducing a negative number from numerical underflow.
Because our data is large, this is the approach that we use.
As was mentioned previously,
is sparse, which facilitates a simpler com-
putation in the procedure above. The matrix Z from each step should not
be formed explicitly because it would be a large, dense matrix. Instead, the
product of a matricized
X
with Z should be computed specially, exploiting the
inherent Kronecker product structure in Z so that only the required elements
in Z need to be computed and multiplied with the nonzero elements of
X
X
.
See ( 3 ) for details.
5.4 Enron Subset
The original collection of Enron emails used in this study (and in the NTF
discussed in (5)) is available online (12). Although this collection comprises
517,431 emails extracted from 150 different mail directories, we use the Enron
email subset (or graph) prepared by Priebe et al. (30) that consists of messages
among 184 Enron email addresses plus thirteen more that have been identified
in (6) as interesting. We considered messages only in 2001, which resulted in
a total of 53 , 733 messages over 12 months (messages were sent on a total of
357 days).
As discussed in (5), the lack of information on the former Enron employees
has hampered the performance evaluation of any model of the Enron Email
Set. Having access to a corporate directory or organizational chart of Enron
at the time of these emails (at least for the year 2001) would greatly help test
the validity of results (via PARAFAC or any other model). Other researchers
using the Enron Email Set have had this same problem. Hopefully, in time,
more historical information will be available. Illustrations of the true/false
 
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