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Congratualtions to the winner of the Regular Equivalence contest, is Martin Everett. As you may recall, 3 or 4 years ago I offered a \$100 for any naturally occurring regular equivalence that was significant at the .05 level, as calculated according to my paper with Kai Jonas:

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

In this paper we calculated the probability of a 1-block being regular, conditioned on the number of 1s in the block. If you think about this for a minute, it means that a block of all 1s has to be regular (no zero rows or columns) and so is no evidence for regularity. This is like linear regression on data that is constant. It fits a line perfectly but is not significant because it is explained by the constant model. Anyway, Martin's data (which came somewhat embarrassingly from ucinet under "taro exchange" and is described in Per Hage and Harary's 1991 book, Exchange in Oceania) is given below.

> If this dosen't look good copy and paste it into word and set to courier
> 10
> point.
>
>          1 2 1 1   1 1       1 1 2   1   1       1 2
>      6 2 4 0 0 1 7 3 7   4 1 5 9 1 9 8   2 3 8 5 6 2
>     -------------------------------------------------
>  6 |             1     | 1             |       1     |
>  2 |                 1 |   1           |   1         |
> 14 |               1   |     1         | 1           |
> 20 |           1       |         1     |       1     |
> 10 |           1       |           1   |     1       |
> 11 |       1 1         |         1   1 | 1           |
>  7 | 1                 | 1     1       |     1   1   |
> 13 |     1             |           1   | 1           |
> 17 |   1               |   1         1 |   1     1 1 |
>    ---------------------------------------------------
>  4 | 1           1     |               |   1   1     |
>  1 |   1             1 |               |       1     |
> 15 |     1             |               | 1       1   |
> 19 |             1     |               | 1         1 |
> 21 |       1   1       |               |       1     |
>  9 |         1     1   |               |     1       |
> 18 |           1     1 |               |           1 |
>    ---------------------------------------------------
> 12 |     1     1   1   |     1 1       |             |
>  3 |   1             1 | 1             |             |
>  8 |         1   1     |           1   |             |
>  5 | 1     1           | 1 1     1     |             |
> 16 |             1   1 |     1         |             |
> 22 |                 1 |       1     1 |             |
>     --------------------------------------------------
>
Since it is symmetric I only tested the upper left diagonal block and the three upper right blocks. The number of regular errors (zero rows plus zero columns within these blocks) is 0. which is 5.5 standard deviations less than expected if the some number of 1s were scattered randomly. If we exclude the one diagonal 1-block (because of symmetry) it is still 3.9 sd's regular. This constrasts with all the other data sets that Kai and I have examined, which tended to be way on the other side of regularity (i.e. with more errors in regularity than expected). I haven't been able to get hold of the Hage and Harary book yet, but there must be some sort of regularity norm in the Taro exchange network. So to paraphase Bill Clinton comment about abortion, regularity is legal, but rare.

Although I'm sure that Martin will make good use of the prize money, either for himself or for a charity, but I am now officially ending the offer, although I would appreciate any more example of naturally occurring regular equivalence.

John P. Boyd

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