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.