Table 1 Glossary of terms and concepts sequentially introduced throughout the text, along with a short explanation

Resource-consumer-function (RCF) tensor \({{\mathcal{F}}}=\{{f}_{ix}^{\alpha }\}\)

\({f}_{ix}^{\alpha }\) is the probability of observing a resource plant species i interacting with a consumer species x via a function α. For phytocentrical field-sampling, \({f}_{ix}^{\alpha }={m}_{ix}^{\alpha }/{n}_{i}\), where \({m}_{ix}^{\alpha }\) is the number of annotated occurrences of i interacting with x via α, and n_i is the number of individuals of observed resource species i. \({{\mathcal{F}}}\) is a rank-3 tensor summarizing the whole observational dataset, the choice of covariant and contravariant indices are set for convenience.

Multilayer ecological network

A specific multilayer network visualization of the RCF tensor, where we choose resources i (plant species) and consumers x (animals/fungi) to be the nodes, and functions α to be the layers. The values of \({f}_{ix}^{\alpha }\) provide the weight of the link between i and x at each layer α (such weight is indeed a probability). After color-coding each layer, the multilayer network is visualized as an edge-colored one (Fig. 2). Since within each layer interaction only takes place between i and x (not directly between i and i or x and x), the multilayer network is also bipartite.

Resource-function bipartite network P, with entries \(\{{P}_{i}^{\alpha }\}\) (Fig. 3)

Obtained from the RCF tensor by suitably integrating out the consumer index according to Eq. (1), \({P}_{i}^{\alpha }\) is the probability of observing a resource i participating in a function α. The matrix P is interpreted as the weighted biadjacency matrix of a bipartite resource-function network that accounts for how intertwined resources and functions are within the ecosystem.

Function-function network, with weighted adjacency matrix Φ = P^⊤P and elements {Φ^αβ}

\({\Phi }^{\alpha \beta }={\sum }_{i}{P}_{i}^{\alpha }{P}_{i}^{\beta }\) is the number of different plant species that simultaneously participate in both functions α and β.

Conditioned function-function network Φ∣_i = {Φ^αβ∣_i} with elements \({\Phi }^{\alpha \beta }{| }_{i}:={P}_{i}^{\alpha }{P}_{i}^{\beta }\)

Disaggregation of Φ: for each (resource) plant species i we have a different conditioned function-function network Φ∣_i illustrating how similar functions are in such species embedding.

(Multifunctional) species keystoneness score k_species(i)

\({k}_{{{\rm{species}}}}(i)={\sum }_{\alpha }\left[\frac{{\sum }_{\beta }{\Phi }^{\alpha \beta }{| }_{i}}{{\sum }_{\beta }{\Phi }^{\alpha \beta }}\right]\)

Plant-plant network, with weighted adjacency matrix Π = PP^⊤ and elements {Π_ij}

Its elements \({\Pi }_{ij}={\sum }_{\alpha }{P}_{i}^{\alpha }{P}_{j}^{\alpha }\) quantify the expected number of shared functions by two plant species i and j, or alternatively how similar these two plant species are according to the functions they share.

Conditioned plant-plant network Π∣^α = {Π_ij∣^α} with elements \({\Pi }_{ij}{| }^{\alpha }={P}_{i}^{\alpha }{P}_{j}^{\alpha }\).

Disaggregation of Π: for each ecological function α we have a different conditioned plant-plant network Π∣^α illustrating how similar plant species are in such function embedding.

Function keystoneness score k_function(α)

\({k}_{{{\rm{function}}}}(\alpha )={\sum }_{i}\left[\frac{{\sum }_{j}{\Pi }_{ij}{| }^{\alpha }}{{\sum }_{j}{\Pi }_{ij}}\right]\)