Online Supplementary Material Table S1.Degree centralities for each node in the coffee agroecosystem example, based on the number of edges or hyperedges to which each focal node is incident (i.e. the number of interactions in which the focal species participates). This analogue of degree is calculated as the sum of a focal species' column (or row) of the adjacency matrix, divided by (n-1) where n is the number of species (which normalizes by the maximum degree possible in the graph representation). Web diagrams illustrating these data are included as Online Supplementary Material Figure S1, and the code that generated the data is provided as Online Supplementary Material Code.
Online Supplementary Material Table S2.Degree centralities for each node in the coffee agroecosystem example, based on the number of other nodes to which each focal node is adjacent (i.e. the number of other species with which the focal species interacts). This analogue of degree is calculated as the number of nonzero entries in a focal species' column (or row) of the adjacency matrix, divided by (n-1) where n is the number of species (which normalizes by the maximum degree possible in any of the three representations we compare). Web diagrams illustrating these data are included as Online Supplementary Material Figure S2, and the code that generated the data is provided as Online Supplementary Material Code.
Online Supplementary Material Table S3.Closeness centralities for each node in the coffee agroecosystem example. This metric reflects the focal species' average proximity to other species in the network. The closeness centrality of node i is given by (n-1)(Fi-1), where Fi is the sum of the shortest path distances from the focal node to all other nodes, and n is the number of species. Multiplication by (n-1) normalizes by the maximum possible shortest path distance. Web diagrams illustrating these data are included as Online Supplementary Material Figure S3, and the code that generated the data is provided as Online Supplementary Material Code.
Online Supplementary Material Table S4. Eigenvector centrality data from which Figure 1 a was drawn. Eigenvector centrality was calculated by solving for the eigenvector associated with the largest eigenvalue of the adjacency matrix, with each node's centrality being given by one corresponding element of that eigenvector. The code that generated the data is provided as Online Supplementary Material Code.
Online Supplementary Material Table S5. Betweenness centrality data from which Figure 1 b was drawn. The betweenness centrality of node i is given by [(Sxyi/Txy)][2 ÷ {(n-1)(n-2)}], where Txy is the total number of shortest paths or hyperpaths between nodes x and y, Sxyi is the number of those shortest paths/hyperpaths on which i appears, and the summation is over all pairs of species x and y where xyi. The second term in square brackets normalizes by the maximum possible betweenness in the graph case. Hyperpaths are defined here as in Figure III b and d, and n is the number of species. The code that generated the data is provided as Online Supplementary Material Code.
Online Supplementary Material Figure S1. Illustrations of the data in Online Supplementary Material Table S1. Symbols and colors are as in Figure 1; node sizes and colors are again normalized within each web diagram and not across diagrams.
Online Supplementary Material Figure S2. Illustrations of the data in Online Supplementary Material Table S2. Symbols and colors are as in Figure 1; node sizes and colors are again normalized within each web diagram and not across diagrams.
Online Supplementary Material Figure S3. Illustrations of the data in Online Supplementary Material Table S3. Symbols and colors are as in Figure 1; node sizes and colors are again normalized within each web diagram and not across diagrams.
Online Supplementary Material Figure S4.Potential scheme for handling weakening modifications in aweighted (and undirected) hypergraph. Weights are consistent with each direct interaction having a strength of 100, Azteca reducing the magnitude of the scale:beetle direct interaction by 80%, and phorids in turn reducing the magnitude of that reduction by 80%. Sums of each row of the incidence matrix (I) give the net total weight of interactions in which each species participates, with modifiers being 'given credit' for the magnitude of their modification's effect on a basal direct interaction. The adjacency matrix A is equal to I the transpose of (I 0), with the diagonal elements then subtracted (set equal to zero). Element Aij of the adjacency matrix equals the sum across row i of the incidence matrix of all hyperedges in which jappears (i.e. columns for which the jth row of the incidence matrix contains a nonzero entry). The total net weight of hyperedges in which phorids, Azteca, scales, and beetles participate here (comparable to degree centrality in the unweighted case) is 164, 216, 184, and 84, respectively. If phorids were deleted, those numbers for Azteca, scales, and beetles would become 180, 120, and 20 respectively. Note that this equates to phorid deletion causing an increase in Azteca ants' importance relative to scales, and a decrease in beetles' relative importance. This reflects the fact that the IM imposed by phorids serves to weaken the IM imposed by Azteca ants, which indirectly strengthens the sole direct interaction in which beetles participate.
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