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> On 2015/02/08, at 23:11, Alexander Loewi <[log in to unmask]> wrote:
> 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? 


Re this second question. At the moment my attention is focused on components. Why? It is a useful property of 2-mode networks that the the number of components and their distribution by size in a multi-edge network remain the same whether we simplify the network or project 1-mode networks from it. This follows directly from the fact that, by definition, in 2-mode networks edges can only connect nodes from different modes. 

Since my networks are relatively large ones their large components mostly include around 90% of the nodes. Extracting the large component thus creates a single, fully connected network to use as a starting point. Here is where the fun begins. Removing one of the relationships described in my previous message shatters the original large component into multiple smaller components. But this is not solely a function of that relationship's proportion of lines in the large component. Some relationships with a relatively small number of lines shatter the large component into a relatively large number of new components. Removing relationships with a relatively large number of lines in the large component can result in a smaller number of components. It seems reasonable to suspect that relationships that account for larger numbers of bridging edges will result in more new components, i.e., a larger shatter effect. How to describe mathematically the conditions and scale of that effect is the problem I am pondering.



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