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---
title: Graph embedding and transfer learning can help predict potential species interaction networks despite data limitations
authors:
- name: Tanya Strydom
affiliation:
- ref: udem
- ref: qcbs
orcid: 0000-0001-6067-1349
email: tanya.strydom@umontreal.ca
- name: Salomé Bouskila
affiliation:
- ref: udem
roles: writing
corresponding: false
orcid: 0000-0003-0193-5441
email: salome.bouskila@umontreal.ca
- name: Francis Banville
affiliation:
- ref: udem
- ref: qcbs
- ref: udes
orcid: 0000-0001-9051-0597
email: francis.banville@umontreal.ca
- name: Ceres Barros
affiliation:
- ref: ubcforest
orcid: 0000-0003-4036-977X
email: cbarros@mail.ubc.ca
- name: Dominique Caron
affiliation:
- ref: mcgill
- ref: qcbs
orcid: 0000-0002-2151-6693
email: dominique.caron@mail.mcgill.ca
- name: Maxwell J. Farrell
affiliation:
- ref: uoft
orcid: 0000-0003-0452-6993
email: maxwell.farrell@utoronto.ca
- name: Marie-Josée Fortin
affiliation:
- ref: uoft
orcid: 0000-0002-9935-1366
email: mariejosee.fortin@utoronto.ca
- name: Benjamin Mercier
affiliation:
- ref: udes
- ref: qcbs
orcid: 0000-0002-4104-9463
email: benjamin.b.mercier@usherbrooke.ca
- name: Laura J. Pollock
affiliation:
- ref: mcgill
- ref: qcbs
orcid: 0000-0002-6004-4027
email: laura.pollock@mcgill.ca
- name: Rogini Runghen
affiliation:
- ref: ucecology
orcid: 0000-0001-5785-8321
email: rogini.runghen@gmail.com
- name: Giulio V. Dalla Riva
affiliation:
- ref: ucmath
orcid: 0000-0002-3454-0633
email: giulio.dallariva@canterbury.ac.nz
- name: Timothée Poisot
affiliation:
- ref: udem
- ref: qcbs
orcid: 0000-0002-0735-5184
corresponding: true
email: timothee.poisot@umontreal.ca
affiliations:
- id: udem
name: Département de Sciences Biologiques, Université de Montréal, Montréal, Canada
- id: qcbs
name: Québec Centre for Biodiversity Science, Montréal, Canada
- id: udes
name: Département de Biologie, Université de Sherbrooke, Sherbrooke, Canada
- id: ubcforest
name: Department of Forest Resources Management, University of British Columbia, Vancouver, B.C., Canada
- id: mcgill
name: Department of Biology, McGill University, Montréal, Canada
- id: uoft
name: Department of Ecology & Evolutionary Biology, University of Toronto, Toronto, Canada
- id: ubccons
name: Department of Forest and Conservation Sciences, University of British Columbia, Vancouver, Canada
- id: ucecology
name: Department of Forest and Conservation Sciences, University of British Columbia, Vancouver, Canada
- id: ucmath
name: Department of Forest and Conservation Sciences, University of British Columbia, Vancouver, Canada
bibliography: references.bib
abstract: |
- Metawebs (networks of potential interactions within a species pool) are a powerful abstraction to understand how large-scale species interaction networks are structured.
- Because metawebs are typically expressed at large spatial and taxonomic scales, assembling them is a tedious and costly process; predictive methods can help circumvent the limitations in data deficiencies, by providing a first approximation of metawebs."
- One way to improve our ability to predict metawebs is to maximize available information by using graph embeddings, as opposed to an exhaustive list of species interactions. Graph embedding is an emerging field in machine learning that holds great potential for ecological problems.
- Here, we outline how the challenges associated with inferring metawebs line-up with the advantages of graph embeddings; followed by a discussion as to how the choice of the species pool has consequences on the reconstructed network, specifically as to the role of human-made (or arbitrarily assigned) boundaries and how these may influence ecological hypotheses.
---
# Introduction
The ability to infer potential biotic interactions could serve as a significant breakthrough in our ability to conceptualize networks over large spatial scales [@Hortal2015Seven]. Reliable inferences would not only boost our understanding of the structure of species interaction networks, but also increase the amount of information that can be used for biodiversity management. In a recent overview of the field of ecological network prediction, @Strydom2021Roadmap identified two challenges of interest to the prediction of interactions at large scales. First, there is a relative scarcity of relevant data in most places globally -- which, due to the limitations in most predictive methods, restricts the ability to infer interactions to locations where it is least required (*i.e.* regions where we already have interaction data) leaving us unable to make inference in data scarce regions (where we most need it); second, accurate predictors are important for accurate predictions, and the lack of methods that can leverage a small amount of *accurate* data is a serious impediment to our predictive ability. In this contribution, we (i) highlight the power of viewing (and constructing) metawebs as *probabilistic* objects in the context of low-probability interactions, (ii) discuss how a family of machine learning tools (graph embeddings and transfer learning) can be used to overcome data limitations to metaweb inference, and (iii) highlight how the use of metawebs introduces important questions for the field of network ecology.
In most places, our most reliable biodiversity knowledge is that of a species pool where a set of potentially interacting species in a given area could occur: through the analysis of databases like the Global Biodiversity Information Facility (GBIF) or the International Union for the Conservation of Nature (IUCN), it is possible to construct a list of species for a region of interest. Following the definition of @Dunne2006Network, a metaweb is the ecological network analogue to the species pool; specifically, it inventories all *potential* interactions between species for a spatially delimited area (and so captures the $\gamma$ diversity of interactions as per @Poisot2012Dissimilarity). However, inferring the potential interactions between these species still remains a challenge. And yet, the metaweb holds valuable ecological information: it represents the joint effect of functional, phylogenetic, and macroecological processes [@Morales-Castilla2015Inferring; @carlson2022; @morales-castilla2021]. Specifically, it represents the "upper bounds" on what the composition of the local networks, given a local species pool, can be [see *e.g.* @McLeod2021Sampling]; this information can help evaluate the ability of ecological assemblages to withstand the effects of, for example, climate change [@Fricke2022Effects]. These local networks may be reconstructed given an appropriate knowledge of local species composition and provide information on the structure of networks at finer spatial scales. This has been done for example for tree-galler-parasitoid systems [@Gravel2018Bringing], fish trophic interactions [@Albouy2019Marine], terrestrial tetrapod trophic interactions [@Braga2019Spatial; @OConnor2020Unveiling], and crop-pest networks [@Grunig2020Crop].
The metaweb itself is not a prediction of local networks at specific locations within the spatial area it covers: it will have a different structure, notably by having a larger connectance [see *e.g.* @Wood2015Effects] and complexity [see *e.g.* @Galiana2022Ecological], than any of these local networks. Local networks (which capture the $\alpha$ diversity of interactions) are a subset of the metaweb's species and its realized interactions, and have been called "metaweb realizations" [@Poisot2015Species]. Differences between local networks and their metawebs are due to chance, species abundance and co-occurrence, local environmental conditions, and local distribution of functional traits, among others. Specifically, although co-occurrence can be driven by interactions [@Cazelles2016Theory], co-occurrence alone is not a predictor of interactions [@Blanchet2020Cooccurrence; @Thurman2019Testing], and therefore the lack of co-occurrence cannot be used to infer the lack of a feasible interaction. Yet, recent results by @Saravia2021Ecological strongly suggested that local (metaweb) realizations only respond weakly to local conditions: instead, they reflect constraints inherited by the structure of their metaweb. This sets up the core goal of predictive network ecology as the prediction of metaweb structure, as it is required to accurately produce downscaled, local predictions.
# A metaweb is an inherently probabilistic object
Treating interactions as probabilistic (as opposed to binary) events is a more nuanced and realistic way to represent them. @Dallas2017Predicting suggested that most interactions (links) in ecological networks are cryptic, *i.e.* uncommon or hard to observe. This argument echoes @Jordano2016Sampling: sampling ecological interactions is difficult because it requires first the joint observation of two species, and then the observation of their interaction. In addition, it is generally expected that weak or rare interactions will be more prevalent in networks than common or strong interactions [@Csermely2004Strong]; this is notably the case in food chains, wherein many weaker interactions are key to the stability of a system [@Neutel2002Stability]. In the light of these observations, we expect to see an over-representation of low-probability (hereafter rare) interactions under a model that accurately predicts interaction probabilities.
Yet, the original metaweb definition, and indeed most past uses of metawebs, was based on the presence/absence of interactions. Moving towards *probabilistic* metawebs, by representing interactions as Bernoulli events [see *e.g.* @Poisot2016Structure], offers the opportunity to weigh these rare interactions appropriately. The inherent plasticity of interactions is important to capture: there have been documented instances of food webs undergoing rapid collapse/recovery cycles over short periods of time [*e.g.* @Pedersen2017Signatures]. Furthermore, because the structure of the metaweb cannot be known in advance, it is important to rely on predictive tools that do not assume a specific network topology for link prediction [@Gaucher2021Outlier], but are able to work on generalizations of the network that capture statistical processes giving it its structure. These considerations emphasize why metaweb predictions should focus on quantitative (preferentially probabilistic) predictions, and this should constrain the suite of models that are appropriate for prediction. Binary classifiers based on probabilities have an extremely robust methodology to validate them, and this applies naturally to the prediction of interactions [@poisot2023].
It is important to recall that a metaweb is intended as a catalogue of all potential (feasible) interactions, which is then filtered for a given application [@Morales-Castilla2015Inferring]. It is therefore important to separate the interactions that happen "almost surely" (repeated observational data), "almost never" (repeated lack of evidence *or* evidence that the link is forbidden through *e.g.* trait mis-match), and interactions with a probability that lays somewhere in between [@Catchen2023Missing]. Although metawebs can (and in practice likely do) include false positives, these are statistically negligible compared to the false negatives. Furthermore, @Strydom2022Food shows that t-SVD embedding is extremely robust to (and able to detect) the presence of false positives. In a sense, because most ecological interactions are elusive, we should consider the direct consequences this has on sampling: once the common interactions are documented, the effort required in documenting each rare interaction will increase exponentially [@Jordano2016Sampling]. Recent proposals in other fields relying on machine learning approaches emphasize the idea that algorithms meant to predict, through the assumption that they approximate the process generating the data, can also act as data generators [@Hoffmann2019Machine]. High quality observational data can be used to infer core rules underpinning network structure, and be supplemented with synthetic data coming from predictive models trained on them, thereby increasing the volume of information available for analysis. Indeed, @Strydom2021Roadmap suggested that knowing the metaweb may render the prediction of local networks easier, because it fixes an "upper bound" on which interactions can exist. In this context, a probabilistic metaweb represents an aggregation of informative priors on the biological feasibility of interactions, which is usually hard to obtain yet has possibly the most potential to boost our predictive ability of local networks [@Bartomeus2013Understanding; @Bartomeus2016Common]. This would represent a departure from simple rules expressed at the network scale [*e.g.* @Williams2000Simple] to a view of network prediction based on learning the rules that underpin interactions *and* their variability [@Gupta2022Simultaneously].
![The embedding process (**A**) can help to identify links (interactions) that may have been missed within the original community (represented by the orange dashed arrows, **B**). Transfer learning (**D**) allows for the prediction of links (interactions) even when novel species (**C**) are included alongside the original community. This is achieved with other ecologically relevant predictors (*e.g.* traits) in conjunction with the known interactions to infer latent values (**E**). Ultimately this allows us to predict links (interactions) for species external from the original sample (blue dashed arrows) as well as missing within sample links (**F**). Within this context the predicted (and original) networks as well as the ecological predictors used (green boxes) are products that can be quantified through measurements in the field, whereas the embedded as well as imputed matrices (purple box) are representative of a decomposition of the interaction matrices onto the embedding space]
[The embedding process (**A**) can help to identify links (interactions) that may have been missed within the original community (represented by the orange dashed arrows, **B**). Transfer learning (**D**) allows for the prediction of links (interactions) even when novel species (**C**) are included alongside the original community. This is achieved with other ecologically relevant predictors (*e.g.* traits) in conjunction with the known interactions to infer latent values (**E**). Ultimately this allows us to predict links (interactions) for species external from the original sample (blue dashed arrows) as well as missing within sample links (**F**). Within this context the predicted (and original) networks as well as the ecological predictors used (green boxes) are products that can be quantified through measurements in the field, whereas the embedded as well as imputed matrices (purple box) are representative of a decomposition of the interaction matrices onto the embedding space]: figures/conceptual_2.png {#fig-embedding}
# Graph embedding offers promises for the inference of potential interactions
Graph (or network) embedding (@fig-embedding) is a family of machine learning techniques, whose main task is to learn a mapping function from a discrete graph to a continuous domain [@Arsov2019Network; @Chami2022Machine]. Their main goal is to learn a low dimensional vector representation of the graph (embeddings), such that its key properties (*e.g.* local or global structures) are retained in the embedding space [@Yan2005Graph]. The embedding space may, but will not necessarily, have lower dimensionality than the graph. Ecological networks are promising candidates for the routine application of embeddings, as they tend to possess a shared structural backbone [see *e.g.* @BramonMora2018Identifying], which hints at structural invariants in empirical data. Assuming that these structural invariants are common enough, they would dominate the structure of networks, and therefore be adequately captured by the first (lower) dimensions of an embedding, without the need to measure derived aspects of their structure (*e.g.* motifs, paths, modularity, ...).
## Graph embedding produces latent variables (but not traits)
Before moving further, it is important to clarify the epistemic status of node values derived from embeddings: specifically, they are *not* functional traits, and therefore should not be interpreted in terms of effects or responses. As per the framework of @Malaterre2019Functional, these values neither derive from, nor result in, changes in organismal performance, and should therefore not be used to quantify *e.g.* functional diversity. This holds true even when there are correlations between latent values and functional traits: although these enable an ecological discussion of how traits condition the structure of the network, the existence of a statistical relationship does not elevate the latent values to the status of functional traits.
Rather than directly predicting biological rules [see *e.g.* @Pichler2020Machine for an overview], which may be confounded by the sparse nature of graph data, learning embeddings works in the low-dimensional space that maximizes information about the network structure. This approach is further justified by the observation, for example, that the macro-evolutionary history of a network is adequately represented by some graph embeddings [Random dot product graphs (RDPG); see @DallaRiva2016Exploring]. In a recent publication, @Strydom2022Food have used an embedding (based on RDPG) to project a metaweb of trophic interactions between European mammals, and transferred this information to mammals of Canada, using the phylogenetic distance between related clades to infer the values in the latent subspace into which the European metaweb was projected. By performing the RDPG step on re-constructed values, this approach yields a probabilistic trophic metaweb for mammals of Canada based on knowledge of European species, despite a limited ($\approx$ 5%) taxonomic overlap, and illustrates how the values derived from an embedding can be used for prediction without being "traits" of the species they represent.
## Ecological networks are good candidates for embedding
Ecological networks are inherently low-dimensional objects, and can be adequately represented with less than ten dimensions [@Braga2021Phylogenetic; @Eklof2013Dimensionality; @Braga2019Spatial]. Simulation results by @Botella2022Appraisal suggested that there is no dominant method to identify architectural similarities between networks: multiple approaches need to be tested and compared to the network descriptor of interest on a problem-specific basis. This matches previous results on graph embedding, wherein different embedding algorithms yield different network embeddings [@Goyal2018Graph], calling for a careful selection of the problem-specific approach to use. Additionally, @ghasemian2020 suggest that in some cases, nodes embeddings can be outperformed by other methods, re-inforcing the need to thoroughly select the appropriate data analysis technique. In @tbl-methods, we present a selection of common graph and node embedding methods, alongside examples of their use to predict interactions or statistical associations between species. These methods rely largely on linear algebra or pseudo-random walks on graphs. All forms of embeddings presented in @tbl-methods share the common property of summarizing their objects into (sets of) dense feature vectors, that capture the overall network structure, pairwise information on nodes, and emergent aspects of the network, in a compressed way (*i.e.* with some information loss, as we later discuss in the illustration). Node embeddings tend to focus on maintaining pairwise relationships (*i.e.* species interactions), while graph embeddings focus on maintaining the network structure (*i.e.* emergent properties). Nevertheless, some graph embedding techniques [like RDPG, see *e.g.* @Wu2021Maximum] will provide high-quality node-level embeddings while also preserving network structure.
::: {#box-gnn .callout-note}
## Box 1 - Graph Neural Networks
One prominent family of approaches we do not discuss in the present manuscript is Graph Neural Networks [GNN; @Zhou2020Graph]. GNN are, in a sense, a method to embed a graph into a dense subspace, but belong to the family of deep learning methods, which has its own set of practices [see *e.g.* @Goodfellow2016Deep]. An important issue with methods based on deep learning is that, because their parameter space is immense, the sample size of the data fed into them must be similarly large (typically thousands of instances). This is a requirement for the model to converge correctly during training, but this assumption is unlikely to be met given the size of datasets currently available for metawebs (or single time/location species interaction networks). This data volume requirement is mostly absent from the techniques we list below. Furthermore, GNN still have some challenges related to their shallow structure, and concerns related to scalability [see @Gupta2021Graph for a review], which are mostly absent from the methods listed in @tbl-methods. Assuming that the uptake of next-generation biomonitoring techniques does indeed deliver larger datasets on species interactions [@Bohan2017Nextgeneration], there is nevertheless the potential for GNN to become an applicable embedding/predictive technique in the coming years.
:::
Graph embeddings *can* serve as a dimensionality reduction method. For example, RDPG [@Strydom2022Food] and t-SVD [truncated Singular Value Decomposition; @Poisot2021Imputing] typically embed networks using fewer dimensions than the original network [the original network has as many dimensions as species, and as many informative dimensions as trophically unique species; @Strydom2021Svd]. However, this is not necessarily the case -- indeed, one may perform a PCA (a special case of SVD) to project the raw data into a subspace that improves the efficacy of t-SNE [t-distributed stochastic neighbor embedding; @Maaten2009Learning]. There are many dimensionality reductions [@Anowar2021Conceptual] that can be applied to an embedded network should the need for dimensionality reduction (for example for data visualization) arise. In brief, many graph embeddings *can* serve as dimensionality reduction steps, but not all do, neither do all dimensionality reduction methods provide adequate graph embedding capacities. In the next section (and @fig-embedding), we show how the amount of dimensionality reduction can affect the quality of the embedding.
| Method | Object | Technique | Reference | Application |
|---------------|---------------|---------------|---------------|---------------|
| t-SNE | nodes | statistical divergence | @Hinton2002Stochastic | [@Cieslak2020Tdistributed, species-environment responses $^a$] [@Gibb2021Data, host-virus network representation] |
| LINE | nodes | stochastic gradient descent | @Tang2015Line | |
| SDNE | nodes | gradient descent | @Wang2016Structural | |
| node2vec | nodes | stochastic gradient descent | @Grover2016Node2vec | |
| HARP | nodes | meta-strategy | @Chen2017Harp | |
| DMSE | joint nodes | deep neural network | @Chen2017Deep | [@Chen2017Deep, species-environment interactions $^b$] |
| graph2vec | sub-graph | skipgram network | @Narayanan2017Graph2vec | |
| RDPG | graph | SVD | @Young2007Random | [@DallaRiva2016Exploring, trophic interactions] [@Poisot2021Imputing, host-virus network prediction] |
| GLEE | graph | Laplacian eigenmap | @Torres2020Glee | |
| DeepWalk | graph | stochastic gradient descent | @Perozzi2014Deepwalk | [@Wardeh2021Predicting, host-virus interactions] |
| GraphKKE | graph | stochastic differential equation | @Melnyk2020Graphkke | [@Melnyk2020Graphkke, microbiome species associations $^a$] |
| FastEmbed | graph | eigen decomposition | @Ramasamy2015Compressive | |
| PCA | graph | eigen decomposition | @Surendran2013Graph | [@Strydom2021Roadmap, host-parasite interactions] |
| Joint methods | multiple graphs | multiple strategies | @Wang2021Joint | |
: Overview of some common graph embedding approaches, by type of embedded objects, alongside examples of their use in the prediction of species interactions. These methods have not yet been routinely used to predict species interactions; most examples that we identified were either statistical associations, or analogues to joint species distribution models. $^a$: application is concerned with *statistical* interactions, which are not necessarilly direct biotic interactions; $^b$:application is concerned with joint-SDM-like approach, which is also very close to statistical associations as opposed to direct biotic interactions. Given the need to evaluate different methods on a problem-specific basis, the fact that many methods have not been used on network problems is an opportunity for benchmarking and method development. Note that the row for PCA also applies to kernel/probabilistic PCA, which are variations on the more general method of SVD. Note further that t-SNE has been included because it is frequently used to embed graphs, including of species associations/interactions, despite not being strictly speaking, a graph embedding technique [see *e.g.* @Chami2022Machine]. {#tbl-methods}
The popularity of graph embedding techniques in machine learning is more than the search for structural invariants: graphs are discrete objects, and machine learning techniques tend to handle continuous data better. Bringing a sparse graph into a continuous, dense vector space [@Xu2021Understanding] opens up a broader variety of predictive algorithms, notably of the sort that are able to predict events as probabilities [@Murphy2022Probabilistic]. Furthermore, the projection of the graph itself is a representation that can be learned; @Runghen2021Exploiting, for example, used a neural network to learn the embedding of a network in which not all interactions were known, based on the nodes' metadata. This example has many parallels in ecology (see @fig-embedding **C**), in which node metadata can be represented by phylogeny, abundance, or functional traits. Using phylogeny as a source of information assumes (or strives to capture) the action of evolutionary processes on network structure, which at least for networks have been well documented [@Braga2021Phylogenetic; @DallaRiva2016Exploring; @Eklof2016Phylogenetic; @Stouffer2007Evidence; @Stouffer2012Evolutionary]; similarly, the use of functional traits assumes that interactions can be inferred from the knowledge of trait-matching rules, which is similarly well supported in the empirical literature [@Bartomeus2013Understanding; @Bartomeus2016Common; @Goebel2023Body; @Gravel2013Inferring]. Relating this information to an embedding rather than a list of network measures would allow to capture their effect on the more fundamental aspects of network structure; conversely, the absence of a phylogenetic or functional signal may suggest that evolutionary/trait processes are not strong drivers of network structure, therefore opening a new way to perform hypothesis testing.
# An illustration of metaweb embedding
In this section, we illustrate the embedding of a collection of bipartite networks collected by @Hadfield2014Tale, using t-SVD and RDPG. Briefly, an RDPG decomposes a network into two subspaces (left and right), which are matrices that when multiplied give an approximation of the original network. RDPG has the particularly desirable properties of being a graph embedding technique that produces relevant node-level feature vectors, and provides good approximations of graphs with varied structures [@Athreya2017Statistical]. The code to reproduce this example is available as supplementary material [note, for the sake of comparison, that @Strydom2021Roadmap have an example using embedding through PCA followed by prediction using a deep neural network on the same dataset]. The resulting (binary) metaweb $\mathcal{M}$ has 2131 interactions between 206 parasites and 121 hosts, and its adjacency matrix has full rank (*i.e.* it represents a space with 121 dimensions). All analyses were done using Julia [@Bezanson2017Julia] version 1.7.2, *Makie.jl* [@Danisch2021Makie], and *EcologicalNetworks.jl* [@Poisot2019Ecologicalnetworks].
{{< embed notebooks/SupplementaryMaterial.ipynb#fig-illustration-1 >}}
In @fig-illustration-1, we focus on some statistical checks of the embedding. In panel **A**, we show that the averaged $L_2$ loss (*i.e.* the mean of squared errors) between the empirical and reconstructed metaweb decreases when the number of dimensions (rank) of the subspace increases, with an inflection at 39 dimensions (out of 120 initially) according to the finite differences method. As discussed by @Runghen2021Exploiting, there is often a trade-off between the number of dimensions to use (more dimensions are more computationally demanding) and the quality of the representation. In panel **B**, we show the increase in cumulative variance explained at each rank, and visualize that using 39 ranks explains about 70% of the variance in the empirical metaweb. This provides different information from the $L_2$ loss (which is averaged across interactions), as it works on the eigenvalues of the embedding, and therefore captures higher-level features of the network. In panel **C**, we show positions of hosts and parasites on the first two dimensions of the left and right subspaces. Note that these values largely skew negative, because the first dimensions capture the coarse structure of the network: most pairs of species do not interact, and therefore have negative values. Finally in panel **D**, we show the predicted weight (*i.e.* the result of the multiplication of the RDGP subspaces at a rank of 39) as a function of whether the interactions are observed, not-observed, or unknown due to lack of co-occurrence in the original dataset. This reveals that the observed interactions have higher predicted weights, although there is some overlap; the usual approach to identify potential interactions based on this information would be a thresholding analysis, which is outside the scope of this manuscript (and is done in the papers cited in this illustration). Because the values returned from RDPG are not bound to the unit interval, we performed a clamping of the weights to the unit space, showing a one-inflation in documented interactions, and a zero-inflation in other species pairs. Panel **D** specifically shows that species pairs with no documented co-occurrence have weights that are not distinguishable from species pairs with no documented interactions; in other words, looking at the embedding, species that do not co-occur are not easily distinguished from species that do not interact. This suggests that (as befits a host-parasite model) the ability to interact is a strong predictor of co-occurrence.
{{< embed notebooks/SupplementaryMaterial.ipynb#fig-illustration-2 >}}
In @fig-illustration-2, we relate the values of latent variables for hosts to different ecologically-relevant data; we can perform this additional step, because the results presented in @fig-illustration-1 show that we can extract an embedding of the metaweb that captures enough variance to be relevant. Importantly, this is true for both $L_2$ loss (indicating that RDPG is able to capture pairwise processes) and the cumulative variance explained (indicating that RDPG is able to capture network-level structure), which suggests that these approaches may allow to predict interactions *and* network structure. In panel **A**, we show that host with a higher value on the first dimension have fewer parasites. This relates to the body size of hosts in the *PanTHERIA* database [@Jones2009Pantheria], as shown in panel **B**: interestingly, the position on the first axis is only weakly correlated to body mass of the host; this matches well established results showing that body size/mass is not always a direct predictor of parasite richness in terrestrial mammals [@Morand1998Density], a result we observe in panel **C**. Finally, in panel **D**, we can see how different taxonomic families occupy different positions on the first axis, with *e.g.* Sciuridae being biased towards higher values. These results show how we can look for ecological informations in the output of network embeddings, which can further be refined into the selection of predictors for transfer learning.
# The metaweb merges ecological hypotheses and practices
Metaweb inference seeks to provide information about the interactions between species at a large spatial scale, typically a scale large enough to be considered of biogeographic relevance (indeed, many of the examples covered in the introduction span areas larger than a country, some of them global). But as @Herbert1965Dune rightfully pointed out, "\[y\]ou can't draw neat lines around planet-wide problems"; any inference of a metaweb must therefore contend with several novel, interwoven, families of problems. In this section, we outline three that we think are particularly important, and discuss how they may be addressed with subsequent data analysis or simulations, and how they emerge in the specific context of using embeddings; some of these issues are related to the application of these methods at the science-policy interface. Adressing these considerations as part of the methodological discussion is particularly important, as the construction of metawebs can perpetuate legacies of biases in data (Box 2).
## Identifying the properties of the network to embed
If the initial metaweb is too narrow in scope, notably from a taxonomic point of view, the chances decrease of finding another area with enough related species (through phylogenetic relatedness or similarity of functional traits) to make a reliable inference. This is because transfer requires similarity (@fig-embedding). A diagnostic for the lack of similar species would likely be large confidence intervals during estimation of the values in the low-rank space. In other words, the representation of the original graph is difficult to transfer to the new problem. Alternatively, if the initial metaweb is too large (taxonomically), then the resulting embeddings would need to represent interactions between taxonomic groups that are not present in the new location. This would lead to a much higher variance in the starting dataset, and to under-dispersion in the target dataset, resulting in the potential under or over estimation of the strength of new predicted interactions. @Llewelyn2022Predicting provided compelling evidence for these situations by showing that, even at small spatial scales, the transfer of information about interactions becomes more challenging when areas rich with endemic species are considered. The lack of well documented metawebs is currently preventing the development of more concrete guidelines. The question of phylogenetic relatedness and distribution is notably relevant if the metaweb is assembled in an area with mostly endemic species (*e.g.* a system that has undergone recent radiation or that has remained in isolation for a long period of time might not have an analogous system with which to draw knowledge from), and as with every predictive algorithm, there is room for the application of our best ecological judgement. Because this problem relates to distribution of species in the geographic or phylogenetic space, it can certainly be approached through assessing the performance of embedding transfer in simulated starting/target species pools.
## Identifying the scope of the prediction to perform
The area for which we seek to predict the metaweb should determine the species pool on which the embedding is performed. Metawebs can be constructed by assigning interactions in a list of species within specific regions. The upside of this approach is that information relevant for the construction of this dataset is likely to exist, as countries usually set conservation goals at the national level [@Buxton2021Key], and as quantitative instruments are consequently designed to work at these scales [@Turak2017Using]; specific strategies are often enacted at smaller scales, nested within a specific country [@Ray2021Biodiversity]. However, there is no guarantee that these arbitrary boundaries are meaningful. In fact, we do not have a satisfying answer to the question of "where does an ecological network stop?", the answer to which would dictate the spatial span to embed/predict. Recent results by @Martins2022Global suggested that networks are shaped within eco-regions, with abrupt structural transitions from an eco-region to the next. Should this trend hold generally, this would provide an ecologically-relevant scale at which metawebs can be downscaled and predicted. Other solutions could leverage network-area relationships to identify areas in which networks are structurally similar [see *e.g.* @Fortin2021Network; @Galiana2018Spatial; @Galiana2022Ecological]. Both of these solutions require ample pre-existing information about the network in space. Nevertheless, the inclusion of species for which we have data but that are not in the right spatial extent *may* improve the performance of approaches based on embedding and transfer, *if* they increase the similarity between the target and destination network. This proposal can specifically be evaluated by adding nodes to the network to embed, and assessing the performance of predictive models [see *e.g.* @Llewelyn2022Predicting].
## Putting models in their context
Predictive approaches in ecology, regardless of the scale at which they are deployed and the intent of their deployment, originate in the framework that contributed to the ongoing biodiversity crisis [@Adam2014Elephant] and reinforced environmental injustice [@Choudry2013Saving; @Dominguez2020Decolonising]. The risk of embedding this legacy in our models is real, especially when the impact of this legacy on species pools is being increasingly documented. This problem can be addressed by re-framing the way we interact with models, especially when models are intended to support conservation actions. Particularly on territories that were traditionally stewarded by Indigenous people, we must interrogate how predictive approaches and the biases that underpin them can be put to task in accompanying Indigenous principles of land management [@Eichhorn2019Steps; @Nokmaq2021Awakening]. The discussion of "algorithm-in-the-loop" approaches that is now pervasive in the machine learning community provides examples of why this is important. Human-algorithm interactions are notoriously difficult and can yield adverse effects [@Green2019Disparate; @Stevenson2021Algorithmic], suggesting the need to systematically study them for the specific purpose of, here, biodiversity governance. Improving the algorithmic literacy of decision makers is part of the solution [*e.g.* @Lamba2019Deep; @MoseboFernandes2020Machine], as we can reasonably expect that model outputs will be increasingly used to drive policy decisions [@Weiskopf2022Increasing]. Our discussion of these approaches need to go beyond the technical and statistical, and into the governance consequences they can have. To embed data also embeds historical and contemporary biases that acted on these data, both because they shaped the ecological processes generating them (see Box 2), and the global processes leading to their measurement and publication. For a domain as vast as species interaction networks, these biases exist at multiple scales along the way, and a challenge for prediction is not only to develop (or adopt) new quantitative tools, but to assess the behavior of these tools in the proper context.
# Conclusion
Although promising, the application of embeddings to metaweb prediction still involved several challenges. First, there is a need to understand how to define a metaweb as a single, cohesive, unit of ecological organisation. This is likely to have very different answers based on the specific taxonomic group, temporal and spatial resolution, and question being investigated. Second, there is a need to understand the scale at which these predictions are relevant. Although we have documented many cases of using embedding to fill gaps in the metaweb, these techniques can likely be brought into a spatial (and possibly temporal) context. The validation of these predictions will have to proceed jointly with empirical sampling of interactions, but also with the design of downsampling methods. Finally, there is a need for a greater understanding of how biases in the data propagate to the predictions. Because the volume of metawebs is currently low, and because graph embeddings have not been commonly applied, we anticipate that this discussion will take place organically in the coming years.
::: {#sec-box-legacy .callout-note}
## Box 2 - Minding legacies shaping ecological datasets
In large parts of the world, boundaries that delineate geographic regions are a legacy of settler colonialism, which drives global disparity in capacity to collect and publish ecological data. Applying any embedding to biased data does not debias them, but rather embeds these biases, propagating them to the models using embeddings to make predictions. Furthermore, the use of ecological data itself is not an apolitical act [@Nost2021Political]: data infrastructures tend to be designed to answer questions within national boundaries (therefore placing contingencies on what is available to be embedded), their use often drawing upon, and reinforcing, territorial statecraft [see *e.g.* @Barrett2005Environment]. As per @Machen2021Thinking, these biases are particularly important to consider when knowledge generated algorithmically is used to supplement or replace human decision-making, especially for governance (*e.g.* enacting conservation decisions on the basis of model prediction). As information on networks is increasingly leveraged for conservation actions [see *e.g.* @Eero2021Use; @Naman2022Food; @Stier2017Integrating], the need to appraise and correct biases that are unwittingly propagated to algorithms when embedded from the original data is immense. These considerations are even more urgent in the specific context of biodiversity data. Long-term colonial legacies still shape taxonomic composition to this day [@Lenzner2022Naturalized; @Raja2022Colonialism], and much shorter-term changes in taxonomic and genetic richness of wildlife emerged through environmental racism [@Schmidt2022Systemic]. Thus, the set of species found at a specific location is not only as the result of a response to ecological processes separate from human influence, but also the result of human-environment interaction as well as the results of legislative/political histories.
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**Acknowledgements:** We acknowledge that this study was conducted on land within the traditional unceded territory of the Saint Lawrence Iroquoian, Anishinabewaki, Mohawk, Huron-Wendat, and Omàmiwininiwak nations. We thank Victoria Hemming for feedback on this manuscript. TP, TS, DC, and LP received funding from the Canadian Institute for Ecology & Evolution. FB is funded by the Institute for Data Valorization (IVADO). TS, SB, and TP are funded by a donation from the Courtois Foundation. M-JF acknowledges funding from NSERC Discovery Grant and NSERC CRC. RR is funded by New Zealand's Biological Heritage Ngā Koiora Tuku Iho National Science Challenge, administered by New Zealand Ministry of Business, Innovation, and Employment. BM is funded by the NSERC Alexander Graham Bell Canada Graduate Scholarship and the FRQNT master's scholarship. LP acknowledges funding from NSERC Discovery Grant (NSERC RGPIN-2019-05771). TP acknowledges financial support from the Fondation Courtois, and NSERC through the Discovery Grants and Discovery Accelerator Supplement programs. MJF is supported by an NSERC PDF and an RBC Post-Doctoral Fellowship.
**Data availability:** The code to collect the data used in figures is distributed as part of the Supplementary Material.
**Conflict of interest:** The authors have no conflict of interests to disclose
**Authors' contributions:** Tanya Strydom and Timothée Poisot conceived the ideas discussed in the manuscript. All authors contributed to writing and editing the manuscript and gave final approval for publication.
# References