# Network equivalence

26 Thursday Feb 2009

Posted by in CEG theory, NSF proposal

≈ 3 Comments

The objective here is to establish the degree of similarity between successive communities, in spite of changing taxonomic compositions and diversities, and ecological/guild diversity.

Let $\mathcal{U}$ represent an ecosystem over time (e.g. the Karoo series) comprising a series of chronologically successive communities, $\mathcal{U}=\{ U_{a},U_{b},\ldots ,U_{n}\}.$ The metanetwork representation of $\mathcal{U}$ is identical for all $U$ if guild diversities are omitted. That is, the unparameterized metanetworks are automorphisms $\forall U.$ Species-level networks (slns) are generated from the parameterized metanetworks, a finite set for each one. The question being addressed here is, how many of those slns are isomorphic between metanetworks? In other words, how many of a community’s networks are identical to networks in the preceding and succeeding communities? This is a very important question from the perspective of CEG dynamics, because in the model equivalence of ecological dynamics can transcend taxonomic composition and identity. Consider the two slns in the first figure. Specifically they are different, but dynamically they will respond identically to perturbation. This isomorphism extends to the guild level also. Imagine for a moment that the species in the figure are actually different guilds, and that guilds 2 and 3 are very different organisms. IF guild diversities permit the generation of slns with the same numbers of links, then isomorphic slns will be generated. Furthermore, we can remove any guild and species identity completely from the networks and maintain an equivalence of CEG dynamics under the following condition: If the perturbed species/nodes are part of a connected subgraph, and the connected graph is isomorphic with another subgraph, then CEG dynamics of the two networks will be identical! A minimum measure, therefore, of the continuity of ecosystem dynamics between successive communities, is the number of slns that are isomorphic between the sln sets.

Unfortunately, determining whether two graphs or networks are isomorphic is an NP-complete problem, and slns are very complex graphs. Given the size of the sln sets for each community, it would be impossible to determine the power of their intersection. Given that CEG dynamics are drawn however from the likeliest region of the sln space, that is, stochastic draws are made from defined trophic link distributions, we are really interested in establishing isomorphism of those subsets, not the entire sln space. So, one procedure would be to generate a set of high probability slns from each community, and then test those for isomorphism. The test would have to be one of elimination, i.e. whittling down the number that could be isomorphic, without ever actually arriving at the number that are isomorphic. But that would give us an upper limit of the number that could be isomorphic, and an upper limit on the measure of similarity between the two communities. In the next post I will outline an MCMC approach to generate sets of slns with high likelihood.

It is entirely possible that the result will be that no networks are isomorphic between communities. In that case, it would be obvious that, given similarity of CEG dynamics, the networks are “close enough”. We would then be faced with the even more difficult task of measuring minimum distances between networks, but this is not at all an impossible task. We’d just need some reasonably powerful computers, and acknowledge the fact that our carbon footprints would far outweigh any benefits to be gained from our work.

# Trophic network probability

07 Saturday Feb 2009

Posted by in CEG theory

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.

# Another view

05 Thursday Feb 2009

Posted by in Visualization

Tags

food webs, modeling, Network theory, networks

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.

# San Francisco Bay community food web

04 Wednesday Feb 2009

Posted by in Visualization

≈ 4 Comments

Tags

connectance, food webs, networks

Now that’s complex! This is a rendering of the metanetwork for the San Francisco Bay food web. The network consists of 163 nodes, each node being a guild. In total, they represent ~1,600 species of invertebrates and fish, as well as four nodes representing various types of autotrophic producers. There are 5,024 links or trophic interactions between the guilds. The dataset currently excludes birds and marine mammals. Those data are being incorporated even as I type! So, when faced with this level of complexity, how does one determine if the system is resilient, or vulnerable to the removal or addition of specific types of species, or can withstand the effects of climate change?

The figure was produced by one of my graduate students, Rachel Hertog, who has done a tremendous amount of work on this project, as well as the Dominican Republican paleocommunities. The data come almost entirely from the collections of the California Academy of Sciences, notably the Dept. of Invertebrate Zoology & Geology, and the Dept. of Ichthyology.

# the written word

01 Sunday Feb 2009

Posted by in Uncategorized

Check out Wordle. It’s fun.
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