2.3. Hypothesis 1 – Effect of water availability on the structure of ant-plant networks
To evaluate if ant-plant interactions become more specialized and modular as drier the environment, we used binary bi-adjacency matrices describing the networks reported in each study. In these matrices, the ant-plant interactions were described as \(a_{\text{ij}}=1\), when the ant species i interacted with the plant species j , and as\(a_{\text{ij}}=0\) otherwise. Among the studies included in our database, 69% presented the frequency with which each ant and plant species interacted with each other (weighted networks). Since this quantitative information was not available in 31% of our networks and that weighted networks were not evenly distributed along our water gradient, we transformed the weighted networks into binary ones (i.e. presence/absence of interaction between each species pair), making all networks comparable among each other.
For each network, we extracted the following metrics: species richness (S), modularity (Q) (Newman 2006), connectance (C) and nestedness (NODF) (Almeida-Neto et al. 2008). The network S is the number of ant and plant species within the network and was calculated by summing the rows and columns of each binary matrix. The modularity (Q) estimates how grouped the interactions are within the networks. This modular pattern arises from the formation of cohesive groups of interacting species (modules) in which species within a given group are more densely connected to each other than to species from other groups (Olesen et al. 2007). The connectance (C) estimates the proportion of realized interactions from the total possible interactions within the network (Blüthgen and Menzel 2006, Blüthgen et al. 2008, Pellissier et al. 2018). The higher the connectance, the higher the proportion of realized interactions between the ant-plant species pairs and, consequently, the more generalized the interaction patterns (Blüthgen et al. 2008). Finally, the nestedness (NODF) describes a pattern in which species with fewer interactions often interact with a subset of the partners interacting with species with a larger number of interactions. Nested networks often show (1) interactions among generalists (species with multiple partners in the network), (2) asymmetries in the number of interactions, in which species with fewer interactions (specialists) often interact with generalists, (3) interactions among specialists are rare or absent (Bascompte et al. 2003, Guimaraes Jr et al. 2006, Bascompte and Jordano 2007a). In a perfect nested network, we expect a continuum in the degree of species specialization with no species segregation into modules of interacting species. Then, if the ant-plant interactions become more specialized and modular as drier the environment, we expect that the ant-plant networks exhibit lower connectivity and nestedness values and higher modularity levels as lower the mean precipitation of the habitat.
When the S, Q, NODF, or C metrics were not directly reported in the original studies, we calculated them using the raw data reported by the authors in the supplementary material of each paper (n = 5) or kindly sent to us by the authors of each study (n = 4; see the acknowledgments). In the case of Q and NODF to obtain a normalized metric (see below), we re-analyze all networks but two networks for the nested metric in Luna et al. (2018). In one case, we could not retrieve all these metrics, but the paper reported the matrix of interaction as a figure (Dáttilo et al. 2014a). In this case, we recreated the matrix of interaction and used it to calculate our metrics of interest. We could not retrieve any focal metrics in six papers from our database. These papers were discarded, and our final database included 13 papers and 63 networks (see the list of all papers used in our analysis in Supporting information). When we retrieved some, but not all metrics of interest from the studies, we excluded the study only from the dataset used to analyze the specific absent metric. Because of it, we had datasets with different sample sizes for different metrics. In one paper, the metrics obtained corresponded to networks containing different types of interaction (e.g. networks containing ant-EFN plants and ant-aphides-plants – Blüthgen and Fiedler 2004a). In this case, we excluded the interactions not involving plants with EFNs and re-analyzed the network formed only by the interactions between plant species with EFNs and ant bodyguards.
When the C values were not directly reported in the studies, we calculated them using the raw data following:
\(C=\frac{I}{\text{PA}}\),
in which I is the number of realized interactions between ant-plant species pairs, P is the number of plant species, andA the number of ant species in the network. To calculate the NODF and the Q values, we used the ANINHADO (Guimarães and Guimarães 2006) and MODULAR software (Marquitti et al. 2013), respectively. To maximize the network‘s modularity, we used the simulated-annealing algorithm (Guimerà and Amaral 2005) available at MODULAR (Marquitti et al. 2013).
Network’s modularity and nestedness can be affected by variation in the network S (Blüthgen et al. 2008). Because of it, direct comparisons of these metrics between different networks may lead to a misleading interpretation of the mechanisms driving the interactions between species pairs (Blüthgen et al. 2008). Hence, after gathering all metrics of interest, we estimated the Z-score of the Q and NODF values. Z-score reduces such bias by standardizing mean Q and NODF values per unit of the standard deviation of these metrics assuming a given theoretical benchmark (z Q and z NODF from now on). To calculate the Z-scores, we calculated the mean and the standard deviation of Q and NODF values for 1000 simulated null networks based on each empirical network in our dataset. To generate the null networks, we used the null model type II (Bascompte et al. 2003), reported at ANINHADO as null model CE, and at MODULAR as null model 2. The null model type II states that the probability of interaction\(\ (P_{\text{ij}})\) between two species (i and j ) is proportional to the degree (number of interactions) of each species. Therefore, the type II null model of an incidence matrix of size P \(X\) A (number of plant species X number of ant species) could be depicted as:
\(P_{\text{ij}}=\frac{1}{2}\left(\frac{k_{i}}{A}+\frac{k_{j}}{P}\right)\),
in which \(k_{i}\) is the degree of each plant species and \(k_{j}\) is the degree of each ant species. Therefore, the Z-score of each metric of interest (NODF or Q) for each network was calculated as:
\(z_{m}=\frac{m_{R}-{\overset{\overline{}}{m}}_{N}}{\sigma_{N}^{2}}\),
in which m is the metric of interest (Q or NODF), \(m_{R}\) is the metric of interest from the real network,\({\overset{\overline{}}{m}}_{N}\) is the mean value of the metric of interest generated by 1000 randomizations by type II null model and\(\sigma_{N}^{2}\) is the standard deviation of the metric of interest in the 1000 null matrices.