“Weighted” nestedness and “classical” nestedness analyses do not measure the same thing in species interaction networks

This post resulted from a question I posed on Twitter last week and hopefully summarises the issue as I see it and the results of the discussion with colleagues that followed.  Let me know if you disagree or if I have missed anything.

The use of network approaches to understanding how plants and their flower visitors interact has revolutionised the study of these and other mutualistic assemblages of species.  It’s a subject I’ve discussed on the blog before, highlighting some of the work we have published – for instance, see Plant-pollinator networks in the tropics: a new review just published and Local and regional specialization in plant–pollinator networks: a new study just published as two recent examples.

One of the recurring patterns that we see in mutualistic species networks (but not in antagonistic ones such as host-parasite and predator prey) is “nestedness”.  In a nested assemblage of species, generalists with lots of links to other species interact with other generalists and with specialists (those species which have few links to other species)Conversely, specialists tend only to link to generalists: specialist-specialist interactions are rare.  In nature, when we rank species in a network from most to least generalised, this sort of relationship looks like this:

South Africa nested

The rows are plants and the columns are pollinators, in this case from an assemblage of asclepiads and their pollinators we studied in South Africa.  A filled cell in the matrix indicates an interaction between that particular plant-pollinator combination.  It’s not perfectly nested by any means, but statistically this is not a random pattern and it comes out as nested when analysed.  There are a few ways of doing this but the most commonly used is the Nestedness metric based on Overlap and Decreasing Fill (NODF) developed by Almeida-Neto et al. (2008).

I first saw nestedness discussed in relation to plant-pollinator interactions in a presentation by Yoko Dupont of her PhD research at a SCAPE meeting in Sweden in 2001.  It was one of those “A-HA!” moments in science when the light bulb switches on and you realise that you are seeing an important new development which adds significant understanding to a field.  Yoko subsequently published her work as Structure of a plant–flower‐visitor network in the high‐altitude sub‐alpine desert of Tenerife, Canary Islands.

The nested pattern of interactions is conceptually derived from earlier work on island biogeography and species-area relationships and was initially developed to apply to interaction networks by Jordi Bascompte and colleagues in Spain and Denmark – see: The nested assembly of plant-animal mutualistic networks.

What was so exciting about this idea to me was that it provided a way to formally analyse what many of us had been observing and discussing for some time: that mutually specialised plant-pollinator interactions between species are rather rare, and that specialists tend to exploit generalists.  This makes perfect sense because specialist-specialist interactions may be more likely to go extinct, though why it does not also apply to host-parasite interactions is far from clear (and in fact the best known specialist-specialist interactions tend to derive from seed parasitism interactions such as fig-fig wasp and yucca-yucca moth relationships).

Fast forward 20 years and the plant-pollinator networks literature has exploded and our methods of analysis are much more sophisticated than they were in the late 1990s and early 2000s.  Every few months researchers are coming up with new ways in which to analyse these networks, mainly using the R environment for statistics and graphing.  Anyone entering the field would be forgiven for being bewildered as to which approaches to use: it’s bewildering enough for those of us who have been following it from the start!

One thing has been particularly bewildering me for a few years now, and that’s the introduction of “weighted” nestedness.  “Weighted” in this sense means that the abundance or interaction frequencies of the species in the network is taken into account in the analyses.  Visually it could look something like this if we code the cells in the network above to represent abundance or frequency (the darker the cell, the more abundant or frequent):

South Africa nested weighted

I’ve just mocked up the network above, it’s not the actual data.  But quite often networks look like this when we weight them: generalist interactions and/or species tend to be more frequent than specialist.  So far, so obvious.  But here’s the thing: networks that are statistically significantly nested when analysed by NODF tend to be not significantly nested when analysed by a new set of weighted metrics such as wNODF or WINE – see the documentation for the bipartite package for details.   And I don’t understand why.  Or rather I don’t understand why we should be using weights in an analysis of nestedness which is, at its heart, an analysis of presence-absence.  Species are either there or they are not, they are either interacting or they are not.  Their frequency or abundance is immaterial to whether a network is nested.  Indeed, assessing frequency of interactions in plant-pollinator networks is fraught with difficulties because (a) there are so many ways in which to do it; and (b) interactions between plants and pollinators in a community can vary HUGELY between years and across the geographical ranges of the species involved.

This should concern the interaction network community because recently I’ve had reviewers and co-authors saying things like: “don’t analyse for nestedness using NODF because wNODF/WINE is The Latest Thing, use that instead”.  But as far as I and the colleagues who commented on Twitter can tell, nestedness and weighted nestedness are different concepts and are not inter-changeable.  Indeed, many of us are struggling to really define exactly what weighted nestedness analyses are actually measuring.  I can define nestedness in simple terms as a verbal concept, without using the word “nested”, as you saw above.  I can’t do that with weighted nestedness, and I have yet to encounter anyone who can.

So the consensus from the Twitter discussion seems to be that:

  • for any study we should use only those analyses that are relevant to the questions we are asking rather than simply running every available analysis because there are lots to choose from.
  • weighted interaction networks that include abundance or frequency are not necessarily superior to binary presence-absence networks.  Again, it depends on the question being asked.
  • we should not treat weighted nestedness as an upgraded or superior version of classical nestedness.  If you are interested in nestedness, use a binary analysis like NODF.

My thanks to the colleagues who contributed to the Twitter discussion:  Nacho Bartomeus, Pedro Jordano, Pedro Luna, Marco Mello, Chris Moore, Timothée Poisot, and Kit Prendergast.  If you want to follow the Twitter discussion, start here:  https://twitter.com/JeffOllerton/status/1159377089319047168


13 thoughts on ““Weighted” nestedness and “classical” nestedness analyses do not measure the same thing in species interaction networks

  1. Marco Mello

    Hi Jeff, thanks for the nice summary! I totally agree with the take-home messages you listed in the end. Indeed nestedness is one of those concepts that generate a lot of confusion in the literature. Specially because many people do not pay attention to the match between the ecological concepts being investigated and the metrics used as their proxies. Let’s develop some interesting points in this discussion.

    First, although binary and weighted nestedness are different concepts, the latter derives from the former, so they are not independent. A network cannot show weighed nestedness, if it doesn’t show binary nestedness. Nevertheless, on the one hand, a dissortative network (with peripheral hubs that dominate particular modules) can show binary nestedness and not weighted nestedness. On the other hand, an assortative network (in which all hubs are strongly connected to one another, forming a core) should have both metrics correlated to one another.

    Second, as already mentioned, which metric to choose depends on the question being asked and, most importantly, on how the testable prediction was deducted from the working hypothesis. I agree with Fründ 2016 et al. (2016 Oikos) that, in many cases, binary metrics are best for representing the fundamental niches of whole species, while weighted metrics fit better as proxies for the realized niches of local populations. But even that should not be taken as a rule of thumb.

    See some ideas about nestedness, modularity, and network topologies in a new paper published by our group: https://doi.org/10.1002/ecy.2796.

  2. naturalistoncall

    Interesting thread – thanks for a posting it for the twitter-averse. It would seem there is a huge potential for sampling bias in counting the numbers of interactions – wrong time, wrong weather, wrong location – to fully represent the entirety of species’ interactions. Perhaps the weighted data more accurately represents a measure of the relative abundance of the interacting species? Logically, the number of interactions would increase if there are few plants and lots of pollinators, and perhaps dip with lots of plants and fewer pollinators. It would be interesting to see a longitudinal study which includes a reliable measure – do any exist? – of relative densities of plants & pollinators. Might this show a lag typical of predator/prey cycles?

    1. jeffollerton Post author

      There aren’t many network studies that run over more than two years and include robust measures of the abundance of organisms as well as their interaction frequency. It’s a serious gap in our knowledge.

  3. Brian McGill

    “for any study we should use only those analyses that are relevant to the questions we are asking rather than simply running every available analysis because there are lots to choose from.”

    Couldn’t agree more, but it’s frustrating that it needs to be said!

  4. Chris Moore

    Thanks for the post, Jeff! I think you captured much of the conversation accurately and eloquently (e.g., the quote from Brian McGill from above).

    Nevertheless, I am still convinced that (1) weighted networks are more realistic representations of ecological systems and (2) the measurement and use of weights will generate more knowledge than binary networks. I should also note that I do agree with one Twitter comment that we have more useful and better vetted binary data available, and note that I recognize what a Herculean effort it takes to collect any of these data.

    A first defense of weighted networks is (1) that they are more realistic representations of ecological systems. I imagine most people agree with this. Binary networks show the presence and absence of an interaction and weighted networks use a proxy that adds a dimension of “interaction realism.” In a perfect world we’d measure fitness (synonymous here with, say, pollinator or seed-disperser effectiveness), but abundance is often more feasible of a field measure.

    Two questions that naturally follow are: (i) how much is too much realism and (ii) what does this mean for nestedness?

    First, (i) I would argue that considering network weights, although adds a layer of complexity, explains significantly more of the ecological system. Imagine if we had, for instance, a plant-herbivore network with highly uneven abundances between 11 species, with 1 species accounting for 99% of individuals in the community and the other 10 species accounting for 0.1% each. Further imagine that all individuals across species consumed the same amount of plant material. If we measured these herbivores’ effects on a plant community, we can imagine a difference between the binary and a weighted matrix. That difference would be that the binary network would give a description of who interacts with whom, but effectively (in terms of fitness), the interactions within the community are all happening directly between the abundant species and the plants they interact with.

    Second, (ii) I think we can extend this thinking of “effective interactions” to nestedness or any other topological metric (e.g., centrality, modularity). Imagine the plant-herbivore example had a binary nested structure and then weighing each edge by abundance. The row or column with the most abundant species would be very heavily weighted, while others would be very lightly weighted. Would this network still be representative of or behave like a nested network? I would guess no. Yes, the other herbivore species have (P/A) interacted with other plants, but have negligible effects on their fitness. One way that I think about it is if you were to have a cut-off weight below which species do not effectively interact: one could imagine that some nested networks would reain a nested structure and some would not. I did this (but I can’t paste images here) with some bipartite networks. I scaled each by the maximum interaction (sum of the weights) then incrementally, by 0.1 proportions, set those above the cut-off proportion as 1 and below 0, like in a binary network, and looked at nestedness. Some networks retained and others lost nested structure.

    [Incidentally, I don’t think the wNDOF is suitable for the way I describe weights above and thought of an interesting way that this could be measured.]

    As a second defense of weighted networks: (2) the measurement and use of weights will generate more knowledge than binary networks. I see a strong analogy to community diversity based on richness alone and diversity based on abundances. Studying richness is certainly very valuable: we have a rich history of the study of richness being a function of area, latitude, or distance from a dispersal source. Studying diversity including abundances is the basis for most theory in population and community ecology. In networks, I understand how binary data generate a description of who interacts with whom and how that’s very valuable. But I think, too, that including meaningful weights will allow us to develop and refine ecological and evolutionary theory, since populations and alleles as measured in frequencies and they interact through weights (e.g., mutualism coefficient in Lotka-Volterra population models, selection coefficients in population genetic models).

    This became long, quickly. I hope this is clear and conveys a defense of weights, which is that I see a lot of value in *both* binary and weighted metrics, but I don’t think that network weights—for nestedness or other metrics—should be rendered useless. Thanks for the discussion :)!

    1. jeffollerton Post author

      Hi Chris – thanks for this, it’s a really useful contribution. I agree, we can get a lot more ecological information out of a network if it’s weighted compared to binary. But we have to know something about the limitations of the weighting and how it might vary between years for the same community: is that 99% herbivore more abundant because it was recorded in an unusual “outbreak” year, for instance. But I’m sure you’d agree with that.

      Your explanation of how you’d see weighted nestedness is an interesting one and certainly makes sense in terms of its computation. But I see two issues with this: (1) the cut off point for weighted interactions is purely arbitrary and there doesn’t seem to me to be any way to make an ecological meaningful decision as to where to set that cut-off. (2) Perhaps more importantly, having a cut off essentially means that we are throwing data away, data that has meaning. Most species in a community are relatively rare, right? And one of the goals of network ecology is understanding how these rare species exist within a community, what their removal might mean for the stability and functioning of a community, etc. Rare species tend to have fewer interactions of course, almost by definition. Of course it depends on the question being asked, but getting rid of those species means we lose a lot of information. In particular we lose insights into how generalist plants support rare pollinators, and how generalist pollinators ensure the reproduction of rare plants. Perhaps for other kinds of interaction networks this is less of a problem, but for plant-pollinator networks these can be crucial questions. Anyway, thanks for your input, it’s proving to be a very stimulating discussion!

      1. Chris Moore

        Indeed, I’m happy to engage because I find this material fascinating!

        I agree that temporal variation is a wrench thrown in any network analysis, wether its intra- or inter-annual variation. Although, it seems that the concerns are valid for both binary and weighted networks if some years some species interact and others they don’t. Or, more practically, when we sample networks, those that are rare one year and abundant the next are more likely to missed and observed as absent and present respectively.

        _Interaction cutoffs_
        Yes, the cutoff/threshold can be arbitrary. I don’t think, however, that’s such a big deal. For example, we can look at species diversity through looking at the effective number of species (Hill’s numbers), which has different weights on commonness and rarity. Another diversity example is q-diversity, where diversity is a function of a parameter, q, that can be viewed continuously weighing rare and common species. I think the same thinking could be extended to interaction diversity. As a hypothetical example, if one wanted to look at network structure across, say, a latitudinal gradient, they could scale certain structural metrics (e.g., complexity, connectance) not by the number of links/interactions, but the number of links/(effective interactions). If the results are different (e.g., Hill’s 0 compared with Hill’s 2), that’s ecologically interesting and meaningful in my opinion!

        [Along these lines, I think the study of relative interaction abundance and using “interaction abundance distributions” and the like would be *really* interesting! I almost submitted a paper on IADs one time, a long time ago.]

        _Data to answer one’s question_
        I strongly agree with your blog and responses that the metric one should use should be what best answers their question. I don’t think ignoring some rare interactions in many cases will yield misleading answers just like I don’t think ignoring interaction effectiveness in many cases will yield misleading answers. Along the lines of stability and functioning of networks, it seems like a lot of work does indeed consider abundance (i.e., the state variables) and effectiveness (e.g., interaction coefficient) while, in most cases, using binary information to decide who interacts with whom. There is of course the work where nodes are removed and the structure changes, but I think stability more reliably comes from dynamic models that use abundance and interaction strength. Binary node removal can give a much better explanation of extinction and structural changes. Nevertheless, in your example of what happens to specialists dependent on a generalist when all individuals of the generalist species are removed, I agree that binary data better answer that that question.

        Thanks again for the engaging discussion!

    2. Rafael Pinheiro

      Chris, I strongly agree with you.
      Just one point that I think is missing in the discussion and that it is related to your conclusion that wNODF is not suitable for your cutoff approach.

      wNODF is not really a fully weighted index. It mixed binary-weighted: weighted overlap, but decreasing FILL (binary). I don’t see it as a measure of weighted nestedness, but rather as a measure of binary nestedness promoted by weighted nestedness.

      That’s why we have proposed a new index, wNODA, which uses the full weighted information. Instead of decreasing fill, it requires decreasing abundances or frequencies (marginal sums). It can deal with situations in which there is weighted nestedness, but it is not reflected in a binary structure, for instance, completely filled matrices.

      I think wNODA would be suitable for your cutoff procedure, but in this case, I don’t know if it would be necessary. Once you are not bound to the binary structure, the very weak interactions tend to have a very weak impact on the metric, and don’t have to be eliminated.

      1. Chris Moore

        Thanks so much for your comment! I hadn’t seen wNODA, but that looks much better measure of weighted nestedness that is independent of fill! Example 1 in your supplement is very compelling. I’ll give this a study, and appreciate bringing it to our attention.

        Also, this seems like something that should definitely be added to the `bipartite` R package: https://github.com/biometry/bipartite I feel like that would be a way for it to be further seen and used.

      2. Rafael Pinheiro

        I am glad you liked it. wNODA is the fully weighted version of NODF.
        Instead of binary overlap, it measures weighted overlap. Instead of decreasing marginal sums on the binary matrix, decreasing marginal sums on the weighted matrix.

        Carsten (author both of the paper and of bipartite) included the function nest.smdm, that calculates also wNODA, in bipartite. However, I don’t know if it is available in the function networklevel, which is the most commonly used for network metrics.

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