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The authors have declared that no competing interests exist.

Conceived and designed the experiments: ELS SA. Performed the experiments: ELS SA. Analyzed the data: ELS. Wrote the paper: ELS SA JTW. Contributed Data: JTW.

The group model is a useful tool to understand broad-scale patterns of interaction in a network, but it has previously been limited in use to food webs, which contain only predator-prey interactions. Natural populations interact with each other in a variety of ways and, although most published ecological networks only include information about a single interaction type (

Ecological interactions are highly diverse even when considering a single species: the species might feed on a first, disperse the seeds of a second, and pollinate a third. Here we extend the group model, a method for identifying broad patterns of interaction across a food web, to networks which contain multiple types of interactions. Using this new method, we ask whether the traditional approach of building a network for each type of interaction (food webs for consumption, pollination webs, seed-dispersal webs, host-parasite webs) can be improved by merging all interaction types in a single network. In particular, we test whether combining different interaction types leads to a better definition of the roles species play in ecological communities. We find that, although having more information necessarily leads to better results, the improvement is only incremental if the linked species remain unchanged. However, including a new interaction type that attaches new species to the network substantially improves performance. This method provides insight into possible implications of merging different types of interactions and allows for the study of coarse-grained structure in any signed network, including ecological interaction webs, gene regulation networks, and social networks.

Networks are a useful tool to understand patterns of interactions in an ecological community. As ecologists have collected more and more network data, the size of published networks has grown dramatically, with many networks now containing hundreds of species. To make sense of these increasingly complex data, we need tools to simplify the network down to its essential structure, allowing us to identify general patterns of interaction in the community.

The group model (equivalent to the stochastic block model from the social science literature, [

A limitation of the group model is the fact that it can only group species based on a single interaction type (usually predator-prey interactions, although it could in principle be applied to any one interaction type). Of course, species in ecological communities interact in diverse ways, and different interaction types operate simultaneously to influence community dynamics [

Here, we extend the group model from unsigned (single interaction type) to signed directed adjacency matrices, allowing ecologists to study the general structure of merged interaction networks. Using this extension of the group model, species in a group tend to interact with other groups

Each row and column represents a species, and each dot in the heatmap represents an interaction between two species (red for negative impact of column on row, blue for positive impact of column on row, white for no interaction). Colors on the outer edge correspond to group membership. In (A), only trophic links are included, and the network is partitioned into 3 groups. In (B), both trophic and nontrophic interactions are included. The mutualism between the light purple and light green groups has caused the green and purple groups from part (A) to split into two subgroups. In this example, nontrophic interactions serve to refine trophic groups into subgroups, but additional interactions could potentially reinforce or directly conflict with groupings based on a single interaction type.

The Makah Tribal Council has granted permission to the Wootton lab for access to Tatoosh Island.

A food web composed of _{ij} is 1 if _{ij} is 1 if the growth rate of

Since we are interested in how species group within an interconnected network, we require that the complete interaction networks are a single weakly connected component (that is, isolated subgraphs were removed).

Interaction data for Tatoosh Island were collected from the intertidal middle zone based on observed interactions and natural history information. This middle zone on Tatoosh is dominated by the mussel

The largest weakly connected component was taken from Doñana Biological Reserve and Norwood Farm (data made available in [

Because taxonomically similar species are generally expected to fill similar roles in a community [

Consider an interaction web with ^{2} −

Now to see this in the context of the group model, consider N when divided into two groups, _{xx}, the probability of a species in group _{xx}, the probability of a link between two species in _{xy}, the probability of a species in _{yx}, _{yy}, _{yy}, _{xy}, and _{yx}, which are defined similarly. Note that _{xy} and _{yx} are not necessarily equal (nor are _{xy} and _{yx}), since _{1} and _{2}, the Bayes factor is given by:
_{i}) is the marginal likelihood

We searched for the optimal grouping using Metropolis-Coupled Markov Chain Monte Carlo (^{3}) with a Gibbs sampler (see

The entropy of a partition

To measure the similarity between two partitions, we then wish to know how much entropy the partitions share. This is known as the mutual information (

Consider the following two partitions for a five-species grouping:
_{ij} is the number of species which are in group _{i⋅} and column totals _{⋅j} are the marginal counts, _{AB} = .102.

Left circle represents _{AB}. All areas are proportional to the values they represent.

Significance of

Both the partitions for the network with all interactions and the network with trophic interactions grouped species in a similar way (

Venn Diagrams showing the similarity between pairs of partitions in the Tatoosh Mussel Bed: (A) the complete and trophic networks, (B) complete and nontrophic networks, (C) trophic and nontrophic, and complete and taxonomic groupings (D and E). Venn Diagrams are structured as in

Alluvial diagrams comparing the species groupings for (A) complete and trophic webs, (B) complete and nontrophic webs, and (C) trophic and nontrophic webs. Complete network coloring matches colors in

The best complete Tatoosh network grouping, displayed in matrix form. Dot colors in the top row and leftmost column represent group identity (19 groups total). Red and blue dots in the matrix are defined as in

The complete grouping was also quite similar to the nontrophic grouping. In contrast to the trophic partition, which captured the general structure of the complete grouping across the entire web (

Plants in the complete Doñana network grouped in a similar way to both the herbivore-removal and mutualist-removal networks. The herbivore-removal and mutualist-removal partitions were much less similar to each other than to the complete partition, although still more similar than expected by chance (Figs

Venn Diagrams for similarity between pairs of plant partitions for the Doñana web: (A) complete and mutualist-removal webs, (B) complete and herbivore-removal webs, (C) mutualist-removal and herbivore-removal webs, and (D) complete network and taxonomic order. Figure structured as in

Alluvial diagrams comparing the plant groupings for (A) complete and herbivore-removal webs, (B) complete and mutualist-removal webs, and (C) herbivore-removal and mutualist-removal webs. All three comparisons show major areas of similarity, but the groupings in (C) have many more conflicts than (A) and (B).

When parasitoids were excluded from the network, results for the Norwood community were qualitatively similar to Doñana. Mutualist-removal and herbivore-removal groupings were similar to the grouping with both mutualists and herbivores (but not parasitoids), but were less similar to each other (Figs

Venn Diagrams for similarity between pairs of plan partitions for the Norwood Farm webs: (A) complete mutualist-removal webs, (B) complete and herbivore-removal webs, (C) complete and parasitoid-removal webs, (D) complete and mutualist-and-parasitoid-removal webs, (E) complete web and taxonomic order, (F) mutualist-removal and herbivore-removal webs, (G) mutualist-removal and mutualist-and-parasitoid-removal webs, (H) herbivore-removal and parasitoid-removal webs, (I) herbivore-removal and mutualist-and-parasitoid-removal webs, and (J) parasitoid-removal and mutualist-and-parasitoid-removal webs. Figure structured as in

Alluvial diagrams comparing the plant groupings for (A) complete and herbivore-removal webs, (B) complete and mutualist-removal webs, (C) herbivore-removal and mutualist-removal webs, and (D) complete and mutualist-and-parasitoid-removal webs. In general, these grouping are more dissimilar than seen in the Tatoosh and Doñana systems, and only (A) and (D) show more similarity than expected by chance.

Including parasitoids in the network markedly changed the resulting group structure. The complete grouping remained similar to the herbivore-removal grouping (which also removes parasitoids, since they only interact with herbivores). However, the mutualist-removal partition was no more similar to the complete one than expected by chance. Surprisingly, the partition for the mutualist-parasitoid-removal was more similar to the complete partition than either the herbivore or mutualist removal groupings.

Taxonomic grouping provided some information about complete groupings for all three networks. The Tatoosh complete grouping is almost perfectly nested within the species classification by kingdom (Figs

Alluvial diagrams comparing complete web groupings with taxonomic groupings for (A) Tatoosh and kingdom, (B) Tatoosh and phylum, (C) Doñana and plant order, and (D) Norwood and plant order. All groupings are more similar than expected by chance. Kingdom matches very closely with the complete Tatoosh grouping, but has so few categories that it still provides very limited information. The other taxonomic groupings have more categories but still provide relatively little information.

The extended group model is able to take large networks of great complexity, with many types of interactions, and condense them down to their essential structure. This results in a significant decrease in network complexity. It is able to reduce the Tatoosh intertidal network from 110 species down to 19 groups of ecologically equivalent taxa. Using a subset of these interaction types reduces the number of groups simply because the model has less information to work with, and indeed we see that the number of groupings in Tatoosh is greater with all interactions than with trophic interactions only (19 and 13 groups, respectively). Thus, using this extension of the group model in conjunction with interaction web information gives us a slightly more refined view of the network structure. It is notable that the Tatoosh groupings corresponded closely to many ecologically natural sets of species. The model does not use any ecological information outside of the network structure itself, but these patterns of interaction alone are enough to make highly specific distinctions, such as between limpets and other types of grazers.

As one possible use of the extended group model, we consider the effects of including or excluding interaction types from a network. In the Tatoosh network, removing interactions did not exclude species from the network, and even removing large numbers of interactions—nontrophic interactions constitute 54% of interactions in this system—had relatively little effect. This means that in these networks, species which have similar patterns of predation also have similar patterns of competition and mutualism, and so forth. In Doñana and Norwood, however, removing interaction types mean that entire classes of species were also included, and these removals had a comparatively large effect on the group structure. This suggests that plants which are similar to mutualists are not necessarily also similar to herbivores.

The grouping differences between these two network types could arise for many reasons. Sampling effects could play a role, since only three networks were available for study. Intrinsic differences between terrestrial and intertidal systems might also have an effect, since marine systems exhibit strong trophic control [

Taxonomic classification provides an obvious natural grouping for species. However, although taxonomic grouping provided some information about the complete group structure (as has been found for food webs in [

The recursive definition of the group can lead to interesting outcomes. For example, parasites have a dramatic effect on Norwood group structure in the absence of mutualists. This is likely the result of a domino effect where parasitoids influence the grouping of herbivores, and herbivores influence the grouping of plants. Thus, when mutualists are removed, parasitoids have a major effect on the broad structure of the system. But in the presence of mutualists, plants are being influenced by both mutualists and herbivores, and the signal is lost. This result adds to the abundant evidence for the importance of including parasites in networks [

The extended group model may help us study and understand interaction networks in a variety of ways. One possible approach is simply to examine the grouping and look for surprises. For example, only crustose and coralline algae form a group separate from other algae based on trophic information in the Tatoosh network, but when nontrophic information is also incorporated, several kelp species form an additional distinct group. This suggests that these two groups interact differently in the network, in a way that specifically relates to their nontrophic interactions. On closer examination of the network structure, this difference is likely related to the fact that these kelps have a negative effect on the growth of the other algal group, but the other algae do not negatively affect the kelps.

Similarly, because the group model identifies ecologically equivalent species, it can be used to identify species which are performing unique roles in the community. In the Tatoosh network, there are three species which are not grouped with any others: detritus, diatoms, and

Another possible application of the group model is to have a simpler version of the network to work with. These simplified networks are easier to take in and comprehend by eye. They may also be useful for finding generalities across networks. This is currently difficult to do, since there are few interaction networks currently available. In the future, it would be interesting to see if communities tend to form similar numbers of groups, if specific species always perform unique roles, if similar groups tend to form at specific trophic levels, and so forth. Since species within a group perform similar roles in the community, we speculate that these species might exhibit similar population dynamics. It is possible that simplifying networks down to their group structure could be a useful way to simplify multi-species dynamical models.

The extended group model is a general method for identifying functionally equivalent nodes in signed directed networks. We have discussed the method as applied to ecological interaction webs, but the methodology could also be used to study the structure of networks of gene regulation [

Supplementary information about networks, taxonomic data, search algorithm, and model comparisons.

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Figure structured as in main text

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Figure structured as in main text

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Table structured as in main text

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Table structured as in main text

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Table structured as in main text

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Species in the Tatoosh mussel bed network are listed in order of grouping in the complete network as shown from left to right in main text

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Thanks to E. Baskerville, G. Barabás, M. Michalska-Smith, C. Pfister, M. Wang, and G. Dwyer for their ideas and suggestions. We are grateful to the Makah Tribal Council for providing access to Tatoosh.