Reading Borgatti and Everett “Two
Algorithms for Computing Regular Equivalence” (1993) about categorical
equivalence, where on page 373 they say:

“A natural measure of the extent of regular
equivalence between a pair of nodes is given by the number of iterations needed
to split them into separate classes.
If they are split after the first iteration it means they have grossly
different relational patterns, since the first iteration essentially splits up
the nodes according to whether they are sinks, sources, or repeaters.”

When working with networks in which there is one or a few
types of relations this makes a lot of sense, because for each relationship
type you can be you can have sink relationship(s) (or not), source relationship(s)
(or not), or combined relationship(s) (or not). So that’s 2^3
=8 possible categories for a single relational type. You’d
only have to differ on one of the three types of relationships to be in a
different category right from the first iteration, but with just one relation
type that might be considered “grossly different”.

But what about
as the number of relationship types involved gets larger? Add a second relationship type and you
have 64 possible categories in the first iteration, right? With several types of relationships, it would, if I understand it correctly, not be difficult for
each node even in a very large network to have a unique category. And two nodes could match perfectly on
say 4 out of 5 relationship types – hard to think of that as “grossly
different” -- or really be grossly different (e.g. not match on any
types), and either way they’d be treated as different categories of
equivalence right from the first iteration.

This would seem to limit the utility of CatRegE
to networks with several relationship types, since nodes that shared somewhat
(but not perfectly) similar relational patterns would be
separated at the start. Or am I miss understanding something?

Blyden Potts