network theory « Roopnarine’s Food Weblog

Jamaican coral reef I

November 6, 2009 proopnarine Leave a comment

Fig. 1 - Species-level trophic link distribution for entire coral reef.

We’ve examined records of fish occurrences on Jamaican reefs for the past 10 years, and compared it to our “master” food web. Of the 196 species in our food web, 136 have records in Jamaica. Many of these species are present in very low numbers, and some reefs are noticeably depauperate, recording less than 60 species. Nevertheless, to be conservative, we assume that we can integrate over all the reefs, thereby counting all 136 species as being present. We next expanded our metanetwork, or guild-level food web (in this case almost exactly the same as a trophic species-based web) to the species level, therefore accounting for all expected links in the food web. For the master or pristine web, this yields an overall connectance of 0.059. The trophic link distribution is shown in Fig. 1. Interestingly, this is clearly not a decay distribution (e.g. power law), but has a definite modality of about 25 links. One needs to question the extent to which under-sampling of natural food webs, and aggregation into trophic species, affects interpretation of link distributions.

The next step of course is to assess the state of the Jamaican reef system. Our initial analysis has been to simply remove the “missing” species (extirpated) from the web, and to re-calculate the statistics. Connectance declines to 0.055. Is this significant? Probably impossible to answer that question for network connectance. Also, it should be noted that hundreds of invertebrate species are included here, and they will dampen the impact of any fish removals or additions. Perhaps the next question regards the link properties of the extirpated species.

Categories: Coral reefs Tags: extinction, networks, food webs, connectance, network theory, coral reef, link distribution

New paper: Ecological modeling of paleocommunity food webs

October 30, 2009 proopnarine Leave a comment

Roopnarine, P. D. 2009. Ecological modeling of paleocommunity food webs. in G. Dietl and K. Flessa, eds., Conservation Paleobiology, The Paleontological Society Papers, 15: 195-220.

Find the paper here:
http://zeus.calacademy.org/roopnarine/Selected_Publications/Roopnarine_09.pdf
or here
http://zeus.calacademy.org/publications/

Power law confirmed

October 25, 2009 proopnarine Leave a comment

Species-level trophic link distribution

Okay, this post just disappeared, so let’s try again. The updated and correct coral reef food web comprises 759 species. The incoming trophic link distribution, when expanded to the species level (compared to the guild level in the previous post), is a definite power law distribution. The log-transformed data (see figure) yield a function of $y = 11196x^{-1.98}$ , i.e. $\gamma=1.98$ . See the earlier coral reef posts to understand why this is significant.

Categories: CEG theory, Coral reefs Tags: networks, food webs, network theory, corals, coral reef, power law

Coral reef trophic levels, & update

October 23, 2009 proopnarine Leave a comment

Guild-level trophic link distribution

Spent a great week at the Annual Meeting of the Geological Society of America. The Paleontology Society session on Conservation Paleobiology was a lot of fun, and my students also presented great posters. Now back to the coral reef.

I’ve been cleaning up the data, because with some much data, errors are bound to creep in. I believe that the current data are now accurate, and the metanetwork statistics are 265 guilds (including primary producers) and 4,651 links. That yields a metanetwork connectance of 0.066. The link distribution should therefore also be different, and indeed it is. The figure shows the no. of links per guild, and the regression plot demonstrates that the distribution is still a power law distribution. The exponent is smaller than previously calculated, ( $\gamma=1.54$ ), but this is the guild-level network and does not reflect species richnesses (yet).

Trophic level vs. no. of links

The next question that I’m looking at is the distribution of trophic levels among guilds and species. I therefore calculated trophic level for all guilds. The first figure (scatter plot) plots trophic level against the number of prey or incoming links to each guild. There are two things to notice: First, the variance of trophic levels decreases as the number of links, or diet generality of the guild increases. Second, the decrease in the variance is asymmetric, in that there is a bias against being a generalist of low trophic level. This is obvious if you look at all the empty space being vacated below the data points as no. of links increases. I can think of two non-exclusive explanations for this. If you think about a food chain, consumers toward the top of the chain simply have more prey to select from (on an evolutionary timescale), and therefore there should be a natural increase in the number of generalists as trophic level increases. Also, note that there are also many specialists of high trophic level. Perhaps the ability to exert power over other species, as a predator, combined with the previous statement, explains this observation. Finally, what is the distribution of trophic levels within the community? The second figure is a simple histogram plot of all non-primary consumer guilds (i.e. omnivores and carnivores). The distribution is approximately normal, with a definite central tendency. On average, most guilds in the reef are of similar trophic level! That’s very interesting. And referring to the previous scatter plot, we know that there is a biased composition in the tails of the distribution, in that the upper tail (higher trophic level) is a mixed composition of specialist to generalist guilds, but the lower tail is basically restricted to low trophic level specialists.

Guild trophic level distribution

Some of you may have noticed that our trophic levels are non-integer numbers. Primary producers all occupy trophic level 1, and primary consumers are trophic level 2. “Above” that, trophic level is calculated on the basis of the trophic levels of your prey. Exactly how we do that will remain a secret for now.

Categories: CEG theory, Coral reefs Tags: coral reef, corals, food webs, network theory, networks, power law, trophic level

Coral reef species link distribution

October 1, 2009 proopnarine Leave a comment

Species-level trophic link distribution.

The data presented in the previous post examined in-link or in-degree distribution at the guild level, i.e. species are aggregated into ecological guilds. A comment on the previous post asked whether we’ve used any grouping algorithms for guild recognition, and the answer is no, at least not yet (and thanks again for the comment). The current guilds are based primarily on trophic habits and habitat, and other features such as the presence of photo- or chemosymbionts. Guild derived algorithmically would be based on species-level network topology, and ideally, the two would be very similar. Anyway, I noticed the comment when I logged on to post the current results. What I’ve done is to expand the guild-level network (metanetwork) to the species-level, and then re-examine the trophic link distribution. There is no guarantee that the two distributions should agree. For example, it is quite possible that guilds of high in-degree (lots of prey), though few in number, are very species rich, and hence one would lose the decay distribution at the species level. Conversely, guilds of low in-degree could be tremendously more species rich, and would expand disproportionately, when compared to high in-degree guilds, when expanded into member species. Nevertheless, for this dataset, when guilds are actually expanded from 255 consumer guilds to 704 consumer species, the scale-free nature of the distribution is reinforced. The new function is y=11158x^-1.981, implying a power law exponent very close to 2. Neat.

Categories: CEG theory, Coral reefs Tags: coral reef, corals, food webs, network theory, networks, power law, real world networks, small world networks

Coral reef food web II

September 30, 2009 proopnarine 2 comments

Trophic link distribution

What sort of network is the coral reef food web? In other words, how are the links or interactions between nodes in a food web distributed? Food webs have been modelled variously as everything from random (Poisson) networks to networks based on exponential, power law or mixed distributions, with or without hierarchical structure. Empirical measures suggest that link distributions in real world food webs follow exponential or power law distributions, perhaps a mixture of both (differentiated by scale). One of my worries with those measures is that they are based on food webs of varying sizes, and more importantly, levels of taxonomic and ecological resolution. So, for example, how much does it matter if your food web covers only a small part of the community’s taxonomic diversity, or only part of the trophic diversity? What about the level of aggregation of species into more inclusive groups? The high resolution of the coral food web presents an opportunity to address some of these questions, and here’s the first one: How are trophic in-links distributed at the guild level? Recall that guilds here are groups of species with potentially the same prey and predators. I say potentially, for while we have very specific trophic data for some species, e.g. heavily studied fish, data are less certain for many smaller or less well known species. Still, there are 265 guilds in this dataset, and 4,756 links (see previous post). The histogram is a basic frequency histogram of the number of links per guild. As predicted on the basis of previously studied food webs, the distribution is a (right-skewed) decay distribution, with a greater number of species possessing fewer prey, i.e. being relative specialists, and a few species having a broad repetoire of prey, i.e. relative generalists. The extreme generalists (to the right or tail of the distribution) are all large sharks, the most extreme being the tiger shark, Galeocerdo cuvier. These species range from microscopic, single-celled dinoflagellates to large carcharhinid sharks!

guild_trophic_link_distrib

What type of distribution is this? A simple logarithmic transform of the data is shown in the second figure, and regression of the data yields the following function: y = 17238x^-1.9496 (r-squared=0.95). The significant and extremely good fit of a linear function to the transformed data suggests that the underlying link distribution is a power law distribution of the form $p(r) = M^{-\gamma}$ , where $p(r)$ is the link probability, $M$ is the number of prey available, and $\gamma$ is the power law exponent. An exponent of ~1.95 is tantalizingly close to other empirical measures. Even more exciting, for me at least, is the fact that we have predicted on the basis of previous work that power law exponents that promote resistance or robustness to secondary extinctions should lie in the range 2-2.5. That work was based on terrestrial food webs from the Late Permian, 250+ million years ago!

Categories: CEG theory, Coral reefs Tags: coral reef, corals, food webs, network theory, networks, power law, real world networks, Robustness, small world networks

Trophic network probability

February 7, 2009 proopnarine Leave a comment

Given any species level topology $(E,V)\text{,}$ where $E$ represents edges or links, and $V$ represents vertices or species, the probability of that specific topology is the probability of the set of interspecific links specified or composing the topology. Therefore, the probability of the network, given metanetwork $U\text{,}$ is

$p(E,V) = \prod_{x=1}^{\sum \vert G_{u}\vert} \prod _{u=1}^{\vert U \vert}p(r_{x}^{u} \mid r_{x})$
where the right product is the probability of any species $x$ having a particular pattern or topology of links to other species, and the left product is the product of those probabilities for all species in the community. We can see immediately how the probability of any particular species-level food web is built from the probabilities of individual species networks.

The number of species-level networks that can be derived from a metanetwork depends upon permutations of all possible combinations of link topologies of species in the community, and is generally an astronomical number for even a modest number of species and guilds. The network with the greatest probability, or maximum likelihood of occurrence, is one where the probabilities in the formula above are maximized. This can be approximated if one considers that the probability of $x_{i}$ being linked to any $x_{j}$ is equal, regardless of the in-degree of $x_{j}\text{.}$ One can therefore consider the probability of linking to any $x_{j}$ to simply be the proportion of the predator’s set of prey species that is represented by $\vert G_{j}\vert\text{.}$ These are simply maximum likelihood estimates of the metanetwork link probabilities.

The in-links of each species are assigned randomly to species in other guilds for which a metanetwork in-link exists, i.e. those other guilds comprise potential prey of the species, and $a_{iu}=1\text{,}$ where the species belongs to guild $i.$ The resulting network may be represented as a $N\times N$ adjacency matrix, $A_{N}\text{,}$ where $N$ is the total number of species in the community. Binary entries $n_{xy}$ indicate whether species $y$ is prey to species $x\text{.}$ Furthermore, row sums, $\sum n_{xy}$ equal the in-degree of species $x.$

The probability of a species $a\text{'s}$ link topology, $p(a_{i})\text{,}$ or the binary pattern of the row $A_{a}\text{,}$ is the product of the probabilities of each link. This can be calculated efficiently as the multinomial probability of a pattern of links spread among the guilds. Say that the probability of a link between $a$ and a species in guild $G_{j}$ is equal to the fraction of the diversity of $a\text{'s}$ prey that is represented by the species richness of $G_{j}\text{,}$ then the probability of $a$ is
$p(a_{i}) = \frac{r_{a}!}{r_{a}^{1}!r_{a}^{2}!,\ldots,r_{a}^{n}!} \prod_{x=1}^{n}\left( \frac{\vert G_{x}\vert}{\vert R_{i}\vert}\right) ^{r_{a}^{x}} \Rightarrow \frac{r_{a}!}{r_{a}^{1}!r_{a}^{2}!,\ldots,r_{a}^{n}!} \left( \frac{1}{\vert R_{i}\vert}\right)^{r_{a}} \prod_{x=1}^{n}\left( \vert G_{x}\vert\right) ^{r_{a}^{x}}$
where $r_{a}^{x}$ is the number of $a\text{'s}$ links to species in guild $G_{x}\text{,}$ $\vert G_{x}\vert$ is the species richness of $G_{x}\text{,}$ and $\vert R_{i}\vert$ is the total number of prey potentially available to $a$ according to metanetwork $U$ , where $a\in G_{i}\text.$ It is important to note that each term in the formula exists iff $a_{ix}=1\text{,}$ i.e. a metanetwork link exists $G_{i}\leftarrow G_{x}.$ This prevents the inclusion of zero probabilities. Finally, the overall probability of the network is calculated as $p(E,V)=\prod_{a=1}^{a=N}p(a)\text{,}$ the product of all species probabilities, which in turn are the products of all link probabilities.

Categories: CEG theory Tags: likelihood, network theory, networks, probability

Another view

February 5, 2009 proopnarine Leave a comment

sf_bay_metanetwork

This is another rendering of the San Francisco Bay food web (see below) using a different drawing algorithm. This view arranges guilds hierarchically instead of in a circular fashion. It is very interesting to note that “layers” roughly equivalent to trophic levels emerge naturally from the data. Primary producer guilds are at the bottom, and top predators at the top!

This figure was rendered by Rachel using AT&T’s Graphviz dot algorithm.

Categories: Visualization Tags: food webs, modeling, network theory, networks

And the answer is…

January 28, 2009 proopnarine Leave a comment

The number of species-level food webs, or variations of the community (such as the web shown in the previous post) that can be derived for the Miocene marine community, comprising 130 heterotrophic species grouped into 25 guilds (plus for autotrophic guilds) is:
$1.354169406609263 \text{x}10^{1213}$

That’s much larger than the paltry $10^{72}$ or $10^{87}$ particles estimated to make up the known Universe! I can feel the weight of my Hindu ancestry here. A very nice discussion of large numbers can be found in The Biggest Numbers in the Universe. Therein is listed an old favourite of mine from The Hitchhiker’s Guide to the Galaxy: “Space is big. Really big. You just won’t believe how vastly hugely mind-bogglingly big it is. I mean, you may think it’s a long way down the road to the chemist, but that’s just peanuts to space.”

So just how meaningful is all the possible community network variation? One caveat to the answer is that primary productivity is specified in the network as units of productivity, and not actual species. So there are actually a total of 1320 nodes representing primary production in the network, and those certainly contribute significantly to the huge number. But let’s say we reduce those nodes to simply one for each producer guild, resulting in substantially smaller species-level networks. The number of possible networks is now
$1.07583\text{x}10^{237}$
That’s still pretty big!

Categories: CEG theory Tags: food webs, network theory, networks

Size of a network space

January 28, 2009 proopnarine Leave a comment

One possible food web for Late Miocene shallow marine community, Dominican Republic.

The metanetwork is a higher guild-level representation of the community food web or trophic network, incorporating uncertainty and variability of the species-level links. The number of species-level networks that can be derived from the network is finite, but large. This species-level network space encompasses all the possible configurations and behaviours that can be exhibited by the community. An interesting excercise is to calculate the size of the space under particular parameterizations of the metanetwork. Let’s do that for the Late Miocene community whose metanetwork is described here. An example of one of the species-level food webs is illustrated in this figure. Okay, here we go…

A metanetwork U is a set of guilds, $\{ G_{i},G_{2}\ldots ,G_{\vert U \vert}\}$ . $\vert U\vert$ represents the power of U, or the number of guilds, and $\vert G_{i}\vert$ is the species-richness or number of species in guild $G_{i}$ . The number of potential prey species for each species in $G_{i}$ is the sum of the species-richnesses of all guilds designated by the topology of U as prey to $G_{i}$ . This number is calculated as

$\vert R_{i}\vert = \sum_{j=1}^{\vert U\vert} a_{ij}\vert G_{j}\vert$

where $\vert R_{i}\vert$ is the number of potential prey species and $a_{ij}$ is the $ij^{th}$ element of U’s binary adjacency or connectivity matrix. The order ij designates i as a predator of j, and $a_{ij}=$ 1 if this is true, and zero otherwise.

If for simplicity we assume that all species in the community are extreme specialists, that is, they consume specifically a single resource, then the number of different species link configurations possible in $G_{i}$ is $\vert R_{i}\vert ^{\vert G_{i}\vert}$ . Therefore in U there are exactly

$\prod_{i=1}^{\vert U\vert} \vert R_{i}\vert ^{\vert G_{i}\vert}$

possible species-level networks. Given that links within each guild are described, however, by a trophic link distribution, and that species within a guild are expected to differ in their dietary breadths, the expected (average) number of species-level networks is found by first calculating the number of possible ways in which the links of a single species may be arranged among potential prey nodes, and the applying that to the above product.

$\prod_{i=1}^{\vert U\vert} \prod_{r_{x}=1}^{\vert R_{i}\vert} \left( \begin{array}{c} \vert R_{i}\vert \\ r_{x} \end{array} \right) ^{\vert x_{i}\vert}$

where $\vert x_{i}\vert$ is the number of species in $G_{i}$ of in-degree (with number of prey) $r_{x}$ . (The formula for calculating $\vert x_{i}\vert$ was given in a previous post.)

Categories: CEG theory Tags: food webs, network theory, networks

Older Entries

Roopnarine’s Food Weblog

Archive