Striving to Map the Shape-Shifting Net

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Internet, real world networks

In a dimly lit chamber festooned with wires and hidden in one of California’s largest data centers, Tim Pozar is changing the shape of the Internet.

Food web complexity and connectance

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connectance, coral reef, food webs, Network theory, networks, paleo-food web, real world networks

Another installation in the series (see previous posts on this page).

System complexity.– The complexity of a food web depends upon the taxon richness of the system, as well as the topology and dynamics of interspecific interactions. Although richness and topology are captured by graphic depictions, the utility of the depictions is often limited to impressing upon the viewer the overwhelming structural complexity of the systems. For example, here is a Greater Antillean coral reef food web comprising 265 trophic guilds and 4,656 interactions, currently one of the most detailed food web networks available. The system is definitely complicated, as expected of a coral reef community, but not much else can be concluded from the graph. In fact, it is more complicated than illustrated, being based on a dataset comprising 750 species and 34,465 interspecific interactions. Many of the species have been aggregated into sets termed trophic guilds, where members of a guild share prey drawn from the same guild(s), and likewise for predators. Species aggregation is a common way in which to reduce food web network complexity, but there are few formulaic methods for aggregation. The most common method is based on the concept of trophic species (trophospecies), where aggregated species are assumed to have exactly the same prey and predators. The trophic guild concept on the other hand was formulated specifically for fossil taxa and assumes uncertainty in species interactions. It is very important to understand the impacts of aggregation on network structure and dynamics, and the implications for species’ roles in the system. Whether different aggregation schemes yield similar insights into complex systems is currently poorly understood. I will return to this topic in a later post.

Connectance.– A number of measures and summary statistics are used to describe and compare food webs, perhaps the most common one being connectance. Food web connectance differs from the graph connectance defined earlier, because the networks are now directional. Each node may link to every other node including itself, but a directional link from species A to B is no longer equivalent to a link from B to A. The maximum number of links possible is therefore the square of the number of nodes. Using symbols common in the food web literature,

where L is the number of directional links in the network, and S is the number of nodes or species. Connectance values are generally well below one, reflecting the relative sparsity of links in food webs, but it is difficult to compare connectances among food webs that use different aggregation schemes. Perhaps given this difficulty, it is quite surprising that there is a regular relationship between L and S spanning a large number of food webs, compiled from a variety of sources, and using different aggregation methods (see also Ings et al.). The exponential nature of the relationship shows that link density, or connectance, increases with increasing node richness. It is possible that increasing taxon richness in a community demands greater connectivity in order to maintain efficient energy transfer and hence stability, or the relationship is simply spurious and any true relationship is obscured by the heterogeneity of food web metadata. This remains, in my opinion, an open problem in food web theory.

Food webs as networks

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connectance, food webs, graph, interaction strength, Network theory, networks, real world networks, Robustness

Perhaps the most obvious structural elements of real food webs that distinguishes them from the graphs presented earlier is directionality of the links. Links are trophic interactions, that is, predator-prey relationships, and describe the passage of energy from prey species to predators. They can also be used to describe the impact of predation on a prey species, recognizing that the relationship is an asymmetrical one between nodes. The “traditional” manner in which to depict this graphically is with arrows between nodes (Fig. A). Whereas the graphs illustrated so far have been undirected graphs, a food web is defined properly as a directed graph, or digraph. The asymmetry is also reflected by the adjacency matrix, which is no longer symmetric about the diagonal.

The most straightforward applications of Graph Theory to food web biology are analyses of the structure or topology of digraphs. Digraphs are often referred to as networks in modern usage, and the study of digraphs, especially those describing real-world networks such as the Internet or social networks, is described as Network Theory. The reader should be aware, however, that networks are technically graphs that are digraphs having weighted or parameterized links. A network therefore depicts a food web when it contains species interactions, the direction of those interactions, and some measure of the interactions, such as interaction strength. A digraph without measures or weights on the links is in reality a special case of a food web digraph, one in which all links are considered equivalent.

A very simple three species food web is illustrated in Fig. A. Species 1 (S1) is prey only (perhaps a primary producer), S2 is both a predator or consumer of S1 while being prey to S3, and S3 is the top consumer in the network. Alternative arrangements for three species are illustrated in Fig. B-D, including a simple food chain (Fig. B), a web where the top consumer is also cannibalistic (Fig. C), and a cycle among the three species (Fig. D). These networks bear only information about the existence and direction of interactions among species, but this information is important because structure always affects function (Strogatz, 2001). The basic network approach has proven useful as a means of capturing the complexity of food webs, deriving basic comparative properties such as connectance and link distributions, and assessing one type of robustness against perturbation.

Graphs and food webs

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food webs, graph, paleo-food web, real world networks

I’m currently working on another review/instructional paper, this one examining the relationship of paleo-food webs to graph and network theory (along with excursions into combinatorics and counting). Here is a draft of one of the sections. This is a draft! No references, and sketches of figures will be added to the post as they are completed.

GRAPHS

A food web is a summary of interspecific trophic interactions. A mathematical graph is the combination of two sets, commonly written G(V, E), where the elements of E are relationships among the elements of V. Both concepts may be expressed graphically as diagrams of relationships among species or elements, an exercise that makes clear the relationship between the real-world biological system and the abstract mathematical one. The area of mathematics dealing with graphs is known as Graph Theory, familiarity with which proves very useful in the exploration and analysis of not only food webs, but of any real-world system (biological and otherwise) that can be expressed as relationships or interactions among discrete entities. Examples of other systems include networks of genomic interactions, metabolic networks, and phylogenetic trees.

Examine the food webs illustrated in Figure 1. The circles represent species, and the links between them are interspecific interactions. Describing these systems mathematically as G(V,E), the elements of E are relationships among the elements of V. The elements of V are typically referred to as vertices or nodes, and their relationships, or the elements of E, are referred to as edges. Edges are written as pairs of vertices, for example , where and are vertices in G (that is, and ). Species in the food web are therefore nodes, and trophic interactions or links are edges. The first web (Fig. 1A) is a system of non-interacting species 1, 2 and 3. It could function only if embedded within a larger system of species with which these species interacted, or if all three species were autotrophic. Such a graph where no vertices or nodes are connected (E is an empty set) is an unconnected graph, one with some edges is connected (Fig. 1B), while a graph with all vertices connected (Fig. 1C) is a complete graph. Any node to which another is linked is termed its neighbor. Note the alternative representation of a graph as an n by n binary adjacency matrix, where element equals one if an edge connects the two vertices, and zero otherwise.

The three graphs would obviously depict food webs with very different implications for the species involved. For example, the density of interactions increases as the number of edges, or |E|, increases. This density is often described simply by the connectance (C) of the graph, standardized as the ratio of the number of edges to the maximum number of edges possible. Each node or species could hypothetically interact with every other species, therefore the number of possible interactions is n(n-1). Note, however, that links would be counted twice, for example {1,2} and {2,1}, so we halve this number. Then

The connectances of Figures 1A-C are therefore zero, 0.333 and 1. We extend this by noting that species may sometimes interact with themselves trophically if individuals are true cannibals. This situation is illustrated in figures as loops, or unit diagonal entries in the adjacency matrix (Fig. 1D). We can now generalize by stating that the connectance of food webs expressed as graphs is measured as

and the connectance of the complete food web in Fig. 1D is therefore 1.

In addition to the overall link density of the graph, or the number of interactions in the food web, we are also interested in the number of interactions per species. This number indicates how trophically specialized or generalized a species is, and is interesting from both evolutionary and ecological perspectives. For example, very specialized species may have stronger coevolutionary interactions with the species to which they are linked, and specialization itself may require temporally extended intervals of stability or high productivity to evolve. Generalized species, on the other hand, could be less susceptible to major perturbations if the intensities of their interactions are distributed broadly among their neighbors. The number of edges or links attached to a node is termed the degree of the node. The simplest cases are those where all nodes are of the same degree, for example Figs. 1A, 1C and 1D. The distribution of links within the graph, or the link distribution, is then single-valued, and may be described as a Dirac delta function or Kronecker’s delta. A slight generalization, where nodes have the same number of links on average, instead of precisely the same degree, leads to the significant development of random graphs and the eventual study of real-world networks.

Next up:Random graphs

Coral reef fish food webs

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coral reef, corals, extinction, food webs, Network theory, real world networks, trophic level

Reef fish food web, Greater Antilles

Here are a couple of renderings of the vertebrate-only component of the coral reef food web. Reminder: the food web is what we expect to see for a reef in the Greater Antilles of the Caribbean, based on data collected around the mid-20th century. The vertebrate component comprises all fish and sea turtle species. The upper figure is the expected food web, and includes 196 species and 995 trophic interactions. Species are arranged on the periphery of the diagram, with interaction represented by the lines crossing the interior. The very busy, or hub species are higher trophic level predators, mostly carcharhinid sharks.

Jamaica coral reef fish food web

The lower figure is what we observe today in Jamaica. (Note: Jamaica is of particular interest for me as a starting comparison, both because of the excellent documentation of those reefs, and my Jamaican heritage; not picking on Jamaica). The number of species, out of 196, observed there over the past 10 years is dramatically smaller. Perhaps more obvious is the loss of interactions. I won’t present the actual data yet, since we will eventually prepare a paper to report all this, but the differences between the two food webs are obvious. We are currently rendering the complete food web, including primary producers and invertebrates, which will be an update of the figures presented in earlier posts. But there are a lot of species in there, and the computers have been churning now for about 17 hours!

New paper: Ecological modeling of paleocommunity food webs

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cascades, competition, connectance, edge strength, extinction, food webs, interaction strength, link strength, modeling, Network theory, networks, nonlinear, paleontology, power law, probability, real world networks, Robustness, Scientific models, simulations, small world networks, Tipping point, top-down cascade

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/

Coral reef species link distribution

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coral reef, corals, food webs, Network theory, networks, power law, real world networks, small world networks

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.

Coral reef food web II

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coral reef, corals, food webs, Network theory, networks, power law, real world networks, Robustness, small world networks

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!

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 , where is the link probability, is the number of prey available, and 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!