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.