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Hi, a few things to clarify: 

-- I'm not entirely sure what you mean by "relationship" here -- an edge of the form artist-writer/art-art/wr-wr? A "boundary-spanner"--"boundary-spanner"? I don't see yet how you could have 10 different kinds with only a 2-mode network -- do the real world examples have more modes?

-- You're looking for a metric that describes the degree of "shatter" for each type of relationship -- but want it to be constant for different densities? Or want it to to be conditional on densities? 

In general, it sounds like you're looking for a "robustness" measure, and that's the literature I'd start with (with which I'm only anecdotally familiar), but I'm just trying to figure out exactly what kind of a statement you're trying to make. I think lots of basic metrics would be quite expressive. For example, "average number of shards produced per edge removed," for each kind of relationship sounds like it would get at what you're talking about pretty successfully -- the biggest problem I can think of with it is that it might not be comparable across networks of different sizes if the nodes are degree-constrained, but that sounds like a pretty secondary concern too. Would that capture what you're looking for?

Best,
Alex.

On Sun, Feb 8, 2015 at 1:59 AM, John McCreery <[log in to unmask]> wrote:
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Dear Colleagues,

On the quantitative side of my research on networks formed by project teams, I encounter the following issue. Members of a team have distinct relationships to the project and thus to each other. Thus, for example, in the advertising industry, the simplest dyadic form of a creative team combines a copywriter with an art director. The former is responsible for words, the latter for the visual image. 

Suppose, for example, that we code these relationships as follows: 1=copywriter, 2=art director. In a 2-mode network of projects whose teams take this form

  • We can imagine a network in which every project has a different copywriter, but some art directors are project boundary spanners who work with more than one copywriter.
  • We can imagine a network in which every project has a different art director, but some copywriters are project boundary spanners who work with more than one art director.
  • Our empirical data show that some copywriter+art director pairs always work together and never work with anyone else but that both copywriters and art directors can be boundary spanners creating links between different projects. 

Our question is how to measure the effects of the two relationships on network properties.

The approach that I have adopted is as follows. Using Pajek, 

  1. I gather basic network information, including average degree, line values, and components.
  2. I extract the large component and gather a second set of basic network information about this subnetwork.
  3. Having converted line values into relations, I remove one of the relationships and note changes in large component subnetwork.

Average degree changes—a predictable consequence of there now being fewer lines in the original network.

The relative proportions of line values also change. Using the hypothetical example described above, in which there are only two relationships, whichever remains now accounts for 100% of line values. In more complex real-world examples, I have up to ten relationships and the changing percentages may tell us something interest.

That said, the most dramatic effect of subtracting one of the relationships is on the large component subnetwork, which now becomes several components, a phenomenon I have labeled “the shatter effect.”  Thus, for example, in my real-world networks, removing one relationship may break the large component into 44 components. Removing another relationship may result in 100 or more. My intuition tells me (1) that this has something to do with the number of edges removed from the original large component and (2) that effects may be disproportionate to the number of edges in question.  I note, for example, that a smaller number of edges, more of which were bridges, might have a larger effect than a larger number of edges, fewer of which are bridges. 

Now, however, I find myself approaching the limits of my mathematical competence. I wonder if anyone has tried a similar approach and, if so, if there are standard ways to measure the disproportionate effects sketched above. 

Your comments and advice will be gratefully received.

John

--
John McCreery
The Word Works, Ltd., Yokohama, JAPAN
Tel. +81-45-314-9324
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http://www.wordworks.jp/
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