Last year, I published a paper about interaction rewiring and how to partition network dissimilarity. From looking at the recent literature, it seems that there is still confusion about this topic (partly due to this paper by Tim Poisot).
The key cause of confusion may be that the purpose for calculating the rewiring component (called betaOS in Poisot’s paper and in my paper) is often not made clear enough. I see mostly two purposes: (case A) you want to quantify the degree of rewiring – then the original definition of Poisot et al. is what you need, or (case B) you want to quantify how much rewiring contributes to total network dissimilarity (implied by wording such as “rewiring drives dissimilarity”) – then the definition based on Novotny is for you. In my research, I am often interested in case A, but my above-mentioned paper focused on case B. Like in other areas of diversity partitioning, the core of the discussion is about denominators and the difference between dissimilarity (A) and additive components of dissimilarity (B).
Let me illustrate the denominator issue with a simple example: Imagine you go out for lunch and there are three options on the menu: Penne Arrabiata (vegetarian pasta), Spaghetti Bolognese (meat pasta) and Pizza Margherita (vegetarian pizza). To calculate the proportion of vegetarian options among pasta, you divide 1 by 2 (the denominator is 2): 50% of the pasta options are vegetarian. The percentage of vegetarian pasta among all options is 33% (the denominator is 3). The percentage of vegetarian meals among all options is 66% (the denominator is 3). All three numbers are informative, but they provide different information. However, you cannot calculate the percentage of vegetarian pizza as 66% – 50% = 16%; it is 66% – 33% = 33%. This shows that percentages are only additive if you use a common denominator, i.e. simple additive operations only work if you stick to percentages in the whole menu.
Back to ecological networks: “meat pasta” has to be replaced by “interactions that are shared between networks”, “vegetarian pasta” has to be replaced by “interactions that differ due to rewiring among species shared between networks”, and “vegetarian pizza” has to be replaced by “interactions that differ due to species turnover (in the wide sense)”. “Percentage” has to be replaced by “dissimilarity”, noting that the dissimilarity measures under consideration are just a special type of percentages.
You have to decide if you are interested in the proportion of pasta that is vegetarian (in the ecological network case: betaOS following Poisot; case A) or in the contribution of vegetarian pasta to all options on the menu (in the ecological network case: betaOS based on Novotny; case B). But I have yet to understand what the number 16% should tell me about the importance of vegetarian pizza on the menu (in the ecological network case: betaST following Poisot, which intends to estimate “the importance of species turnover in the overall dissimilarity”). If there is one vegetarian pasta and one vegetarian pizza, both contribute equally to the number of vegetarian options (in the ecological network case, this holds with betaST based on Novotny) – or would you say that the vegetarian options are mostly pasta and much less pizza (50% vs 16%)? Even beautiful maths proving that a denominator of 2 should be used for the pasta-specific question (case A) wouldn’t convince me that this denominator should also be used for the whole-menu question (case B).
Hope this post helps you think more clearly about what you want when estimating rewiring or partitioning network dissimilarity.
Jochen Fründ 