When are networks worth it?

Last week, during a seminar, my colleague Jesse Shapiro asked a very interesting question: are there some problems for which networks are not “worth it”? Even though my knee-jerk answer was quite obviously “Nah”, the actual answer is far from trivial. So here are a few ways of thinking about it, and absolutely nothing in the way of a solution

The question can be re-framed a little bit – when is it not worth paying the cost that comes with the complexity of dealing with networks? As compared to non-network based analyses, anything that involves networks has a narrower range of measures, requires some more intensive computations to get the answer, and is basically a different way of thinking about objects. So it makes a lot of sense to ask if the increase in explanatory or predictive power justifies adding what are essentially new terms or parameters in our model.

There are two situations, I would argue, where networks will not tell you anything interesting. The first is when you have no interactions, or very close to no interactions. This sounds trivial, but networks are a way of leveraging information about the structure of interactions. Maybe there is a critical graph density below which they don’t really deliver. The second situation is, of course, the opposite. If the network is entirely connected, or close to, then you may as well remove the interactions (unless the strength of interactions is quantified).

So my naive expectation is that networks should be really informative when you have enough interaction to pick up some signal, but not so much that every node is the same. If we play around with the idea of applying Shannon’s entropy to a network, where our two classes are no interaction, and presence of an interaction, it is quite easy to see that the value would peak at density $\rho = 0.5$, i.e. half of the matrix representing our network is filled. We previously found that this density maximizes the variance in the degree distribution and the number of possible network arrangements [@PoisGrav14].

But this is sort of an useless answer, as most (ecological) networks are much less connected than this (@RozdSton01 report that this density is associated with lowest stability in simulated networks, which sounds like a reasonable point to make). The point is, there is no clear cut answer to the question of “How do we know if a network is worth it?”. And reading some recent papers, it is clear that we have a lot of arguments about what networks can deliver, and how to apply measures to extract information, but virtually nothing (that I know of) on when networks should or must be used. This is obviously something to revisit more seriously.