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Last year I awarded Martin Everett my $100 prize for regular equivalence. On closer inspection, however, I find that I have to 
withdraw the honor, but not the money (since it was my error in not checking the claim more carefully and also because I like the fact that he gave it to Amnesty International), of finding the first actual regular equivalence. Here's what was wrong with the data and the partition.

1) When I finally found the source of the data, the Taro exchange relations among 22 people found in UCINET, in the book by Hage and Harary, Exchange in Oceania, the table on page 144 shows that the data was incorrectly reported in UCINET: viz., it was symmetrized by adding 12 new arcs, 10 of which corrected regular errors, defined as zero rows or columns in the blocks induced by the equivalence.

2) The partition was not "naturally occurring" as stipulated for the prize; instead it was found by an optimization heuristic in UCINET. This makes the assumptions of my test invalid, in the same way as sorting a classroom into tall and short students and then doing a t-test for the difference in height. You will almost always find  a "significant difference" but to report it as such would be incorrect. Sometimes there is no a priori partition given with the data, but in this case it was right there in the table, with letters indicating membership in one of three clans.

However, if you do use an optimization program, then one way to assess the statistical significance of the result is to run the program on many permuted copies of the data and compare the results with the original. I did this for the (correct, but binarized) Taro data and found that the original data had 3 regular errors. You can do this using the following UCINET options: 
Network | Roles & Positions | Maximal Regular | Optimization for 3 blocks.  Actually, my first three tries gave me 4 errors, but on 
the fourth, I got the 3, so being generous, let's use the smaller number. Note, however, that UCINET's definition of "error" is regular 
errors plus zero-block errors, whereas my method does not include zero-block errors. It doesn't matter since we are accepting the 
UCINET setup for the sake of a permutation test. Then I ran the same program (but only once each) on 20 randomized (using 
UCINETs routine) versions of the (correct) Taro data with the following number of errors: 3 4 3 2 5 4 4 3 4 5 4 5 3 5 4 5 5 4 5 2. 
Note that there are already 6 cases where there at least as many errors as for the unpermuted data, so it is not necessary to do any more runs to reject the regularity hypothesis at the 5% level. 

Anyone with a copy of UCINET can do this test.  Or my test (which assumes a uniform distributions of zeros and ones within each 
block and computes the number of expected zero rows and columns within each regular block), which has been out for four years:

Boyd, J.P. and Kai J. Jonas. 2001. Are social equivalences ever regular? Permutation and exact tests. Social Networks 23: 87-123.

The question is why haven't social networkers applied SOME test? Regular equivalence is supposed to be a scientific idea subject to falsification, and to refuse to do so is an embarrassment to the field. I hope in the future that the referees and editors of our journals will be more diligent in holding their authors to this basic scientific standard.

The contest will not be reinstated, but I would still like to hear from anyone with a real example of regular equivalences.

yours, John Boyd

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